Semiorthogonal Decompositions and Birational Geometry of Del Pezzo Surfaces Over Arbitrary Fields

SEMIORTHOGONAL DECOMPOSITIONS AND BIRATIONAL GEOMETRY OF DEL PEZZO SURFACES OVER ARBITRARY FIELDS

ASHER AUEL AND MARCELLO BERNARDARA

Abstract. We study the birational properties of geometrically rational surfaces from a derived categorical perspective. In particular, we give a criterion for the rationality of a del Pezzo surface S over an arbitrary ﬁeld, namely, that its derived category decomposes into zero- dimensional components. For deg(S) ≥ 5, we construct explicit semiorthogonal decompositions by subcategories of modules over semisimple algebras arising as endomorphism algebras of vector bundles and we show how to retrieve information about the index of S from Brauer classes and Chern classes associated to these vector bundles.

Introduction In their address to the 2002 International Congress of Mathematicians in Beijing, Bondal and Orlov [31] suggest that the bounded derived category Db(S) of coherent sheaves on a smooth projective variety S could provide new tools to explore the birational geometry of S, in particular via semiorthogonal decompositions. The work of many authors [72], [75], [20], [11], [1] now provides evidence for the usefulness of semiorthogonal decompositions in the birational study of complex projective varieties of dimension at most 4. A survey can be found in [77]. At the same time, the relevance of semiorthogonal decompositions to other areas of algebraic and noncommutative geometry has been growing rapidly, as Kuznetsov [76] points out in his address to the 2014 ICM in Seoul. In this context, based on the classical notion of representability for Chow groups and motives, Bolognesi and the second author [21] proposed the notion of categorical representability (see Definition 1.14), and formulated the following question. Question 1. Is a rational variety always categorically representable in codimension 2? When the base field k is not algebraically closed, the existence of k-rational points on S (being a necessary condition for rationality) is a major open question in arithmetic geometry. We are indebted to H. Esnault, who posed the following question to us in 2009. Question 2. Let S be a smooth projective variety over a field k. Can the bounded derived category Db(S) of coherent sheaves detect the existence of a k-rational point on S? This question, which was the original motivation for the current work, is now central to a growing body of research into arithmetic aspects of the theory of derived categories, see [3], [8], [11], [12], [79], [55], [109], [10]. As an example, if S is a smooth projective surface, Hassett and Tschinkel [55, Lemma 8] prove that the index of S can be recovered from Db(S) as the greatest common divisor of the second Chern classes of objects. Recall that the index ind(S) of a variety S over k is the greatest common divisor of the degrees of closed points of S. These questions also intertwine with concurrent developments in equivariant and descent aspects of triangulated and dg categories in the context of noncommutative geometry, see [7], [47], [74], [97], [105]. In the present text, we study these two questions for geometrically rational surfaces over an arbitrary field k, with special attention paid to the case of del Pezzo surfaces of Picard rank 1. The k-birational classification of such surfaces was achieved by Enriques, Manin [82], and Iskovskikh [57]; the study of arithmetic properties (e.g., existence of rational points, stable rationality, Hasse principle, Chow groups) has been an active area of research since the 1970s, see [38] for a survey. 1 2 ASHER AUEL AND MARCELLO BERNARDARA

One of the most important invariants of a geometrically rational surface S over k is the s Néron–Severi lattice NS(Sks ) = Pic(Sks ) of Sks = S ×k k as a module over the absolute Galois s s group Gk = Gal(k /k), where k is a fixed separable closure of k. The structure of this Galois lattice controls the exceptional lines on S, and hence the possible birational contractions. As the canonical bundle ωS is always defined over k, one often considers the orthogonal complement ⊥ ωS . For del Pezzo surfaces, this lattice is a twist of a semisimple root lattice with the Galois action factoring through the Weyl group, see [70, Thm. 2.12]. C. Vial [109] has recently studied how one can, given a k-exceptional collection on a surface S, obtain information on the Néron– Severi lattice. In particular, he shows that a geometrically rational surface with an exceptional collection of maximal length is rational. In this paper, we deal with the opposite extreme, of surfaces of small Picard rank. Even when S has Picard rank 1, and there are no exceptional curves on S defined over k, our perspective is that the missing information concerning the birational geometry of S can be filled in, to some extent, by considering higher rank vector bundles on S that are generators of canonical components of Db(S). For example, let S be a quadric surface with Pic(S) generated by the hyperplane section OS(1) of degree 2. It is well known that the rationality of S is equivalent to the existence of a rational point. By a result in the theory of quadratic forms, a quadric surface having a rational point is equivalent to the corresponding even Clifford algebra C0 (a quaternion algebra over the quadratic extension defining the rulings of the quadric) being split over its center. Given a rational point x ∈ S(k), the Serre construction yields a rank 2 vector bundle W as an 1 1 extension of Ix by OS(−1). On the other hand, the surface Sks = Pks × Pks carries two rank one spinor bundles, corresponding to O(1, 0) and O(0, 1) (see, e.g., [87]). The vector bundle Wks is isomorphic to O(1, 0) ⊕ O(0, 1), as the point x is the complete intersection of lines in two different rulings. Not assuming the existence of a rational point, such rank 2 vector bundles are only defined étalelocally; the obstruction to their existence is the Brauer class of C0. There do, however, exist naturally defined vector bundles V of rank 4 on S, via descent of W ⊕ W , and satisfying End(V ) = C0. One could say that V “controls” the birational geometry of S in a noncommutative way. Derived categories and their semiorthogonal decompositions provide a natural setting for a noncommutative counterpart of the Néron–Severi lattice with its Galois action. One can P2 i consider the Euler form χ(A, B) = i=0 dim Ext (A, B) on the derived category, and the χ- semiorthogonal systems of simple generators, the so-called exceptional collections. Over ks , these are known to exist and have interesting properties for rational surfaces. In this paper, we consider the descent properties of such collections and show how they indeed give a complete way to interpret the link between vector bundles, semisimple algebras, rationality, and rational points. In particular, on a geometrically rational surface S, the canonical line bundle ωS is ⊥ naturally an element of such a system. Then one can consider the subcategory hωSi , whose object are χ-semiorthogonal to ωS. In practice, it can be more natural to consider the category ⊥ ⊥ AS = hOSi , which is equivalent to hωSi . We describe decompositions (or indecomposability) of this subcategory by explicit descent methods for vector bundles. In this context, certain semisimple k-algebras naturally arise and control the birational geometry of S. These algebras also provide a presentation of the K-theory of S. A sample result of this kind, that does not seem to be contained in the literature, is as follows. Let S be a del Pezzo surface of degree 5 over a field k. It is known that S is uniquely determined by an étalealgebra l of degree 5 over k, see [96, Thm. 3.1.3]. We prove a k- equivalence Db(S) = Db(A), where A is a finite dimensional algebra whose semisimplification is k × k × l. In particular, there is an isomorphism ∼ Ki(S) = Ki(k) × Ki(k) × Ki(l) in algebraic K-theory, for all i ≥ 0. In the context of computing the algebraic K-theory of geometrically rational surfaces, this adds to the work of Quillen [91] for Severi–Brauer surfaces (i.e., del Pezzo surfaces of degree 9), Swan [98] and Panin [88] for quadric surfaces and involution surfaces (i.e., minimal del Pezzo surfaces of degree 8), and Blunk [25] for del Pezzo surfaces of DEL PEZZO SURFACES 3 degree 6. Our method provides a uniform way to compute the algebraic K-theory of all del Pezzo surfaces degree ≥ 5. S. Gille [49] recently studied the Chow motive with integer coefficients of a geometrically rational surface, showing that they are zero dimensional if and only if the surface has a zero cycle of degree one and the Chow group of zero-cycles is torsion free after base changing to the function field. Tabuada (cf. [102]) has shown how the Morita equivalence class of the derived category of a smooth projective variety (with its canonical dg-enhancement) gives the noncommutative motive of the variety. We refrain here from defining the additive, monoidal, and idempotent complete category of noncommutative motives, whose objects are smooth and proper dg categories up to Morita equivalence; the interested reader can refer to [102]. We recall only that any semiorthogonal decomposition gives a splitting of the noncommutative motive (see [101]) and that, as well as Chow motives, noncommutative motives can be defined with coefficients in any ring R. Working with Q-coefficients, there is a well established correspondence between noncommutative and Chow motives, see [103]. Roughly speaking, the category of Chow motives embeds fully and faithfully into the category of noncommutative motives, if one “forgets” the Tate motive. As pointed out in [22], this relation does not hold in general for integer coefficients. In particular, for surfaces, one has to invert 2 and 3 in the coefficient ring to have such a direct comparison (see [22, Cor. 1.6]). We think that a comparison between Gille’s results and those of the present paper could lead to a deep understanding of the different types of geometric information encoded by Chow and noncommutative motives. By a del Pezzo surface S over a field k, we mean a smooth, projective, and geometrically integral surface over k with ample anticanonical class. The degree of S is the self-intersection number of the canonical class ωS. Our main result is the following. Theorem 1. A del Pezzo surface S of Picard rank one over a field k is k-rational if and only if S is categorically representable in dimension 0. In particular, if S has degree d ≥ 5 and any Picard rank, the following are equivalent: (i) S is k-rational. (ii) S has a k-rational point. (iii) S is categorically representable in dimension 0. ⊥ (iv) AS ' hωSi is representable in dimension 0. The equivalence of (i) and (ii) for del Pezzo surfaces of degree ≥ 5 is a result of Manin [82, Thm. 29.4]. Our work is a detailed analysis of how birational properties interact with certain semiorthogonal decompositions. In particular, we study collections (so-called blocks) of completely orthogonal exceptional vector bundles on Sks and their descent to S. The theory of 3-block decompositions due to Karpov and Nogin [64] is indispensable for our work. In our results for del Pezzo surfaces of higher Picard rank, the bound on the degree is quite important. This is not surprising, as del Pezzo surfaces of degree d ≥ 5 have a much simpler geometry and arithmetic than those of lower degree. For example, in smaller degree the existence of a rational point only implies unirationality [82, Thm. 29.4]. On the other hand, a succinct corollary of Theorem1 gives a positive answer to Question1 for all del Pezzo surfaces. Corollary 1. Any rational del Pezzo surface S is categorically representable in dimension 0. This relies on the fact that there is no minimal k-rational del Pezzo surface of degree d ≤ 4 (see [83, Thm. 3.3.1]). So Question2 in smaller degrees, where the existence of a k-point is weaker than categorical representability in dimension 0, remains open. One of the ingredients in the proof of Theorem1 is the description of semiorthogonal decompositions of minimal del Pezzo surfaces of degree ≥ 5. This relies on, and extends, the special cases established by Kuznetsov [71], Blunk [26], Blunk, Sierra, and Smith [27], and the second author [19]. In these cases, it always turns out that there are semisimple k-algebras A1 and A2, Azumaya over their centers, and a semiorthogonal decomposition ⊥ b b (0.1) AS ' hωSi = hD (k, A1), D (k, A2)i 4 ASHER AUEL AND MARCELLO BERNARDARA over k. The algebras Ai arise as endomorphism algebras of vector bundles with special homological features. In particular, over ks , these bundles split into a sum of completely orthogonal exceptional bundles, whose existence was proved by Rudakov and Gorodentsev [52], [92], and by Karpov and Nogin [64]. This generalizes the example of quadric surfaces explained above.

Theorem 2. Let S be a del Pezzo surface of degree d ≥ 5. There exist vector bundles V1 and V2 and a semiorthogonal decomposition b b b b (0.2) AS = hV1,V2i = hD (k, A1), D (k, A2)i = hD (l1/k, α1), D (l2/k, α2)i where Ai = End(Vi) are semisimple k-algebras with centers li, and αi is the class of Ai in Br(li). Finally, S has Picard rank 1 if and only if the vector bundles Vi are indecomposable or, equivalently, the algebras Ai are simple. ⊥ One of our main technical results is that such a decomposition of AS ' hωSi can never occur if S has Picard rank 1 and degree ≤ 4, see Theorem 5.1. In fact, in this case one should expect AS to have a very complicated algebraic description, but not much is known (see Theorem 5.7 and Theorem 5.8). A feature of these semiorthogonal decompositions is that the k-birational class of S is determined by the unordered pair of semisimple algebras (A1,A2) up to pairwise Morita equivalence. As pointed out by Kuznetsov [77, §3], one of most tempting ideas in the theory of semiorthogonal decompositions with a view toward rationality is to define, in any dimension and independently of the base field, a categorical analog of the Griffiths component of the intermediate Jacobian of a complex threefold. Such an analog would be the best candidate for a birational invariant. It turns out that such a component is not well defined in general. However, this is not the case for del Pezzo surfaces. Indeed, we can define the Griffiths–Kuznetsov component GKS of a del Pezzo surface S in the case when S has degree d ≥ 5 or Picard rank 1 as follows. Definition 1. Let S be a minimal del Pezzo surface of degree d over k. We define the Griffiths– Kuznetsov component GKS of S as follows: if AS is representable in dimension 0, set GKS = 0. b If not, GKS is either the product of all indecomposable components of AS of the form D (l, α) with l/k a field extension and α ∈ Br(l) nontrivial or else GKS = AS. 0 If S is not minimal, then we set GKS = GKS0 for a minimal model S → S . Definition1 may appear ad hoc, but it can roughly be rephrased by saying that the Griffiths– Kuznetsov component is the product of all components of AS which are not representable in dimension 0. If deg(S) ≥ 5, then GKS is determined up to equivalence by Brauer classes related to the algebras A1 and A2. If deg(S) ≤ 4 and S has Picard rank 1, we conjecture its indecomposability (see Conjecture 5.6). Theorem 3. Let S be a del Pezzo surface of degree d over k. The Griffiths-Kuznetsov component GKS is well-defined, unless deg(S) = 4 and ind(S) = 1, or deg(S) = 8 and ind(S) = 2. 0 Moreover, if S 99K S is a birational map, then GKS = GKS0 . We notice that the two “bad” cases of degree 4 and index 1 or degree 8 and index 2 are birational to conic bundles which are not forms of Hirzebruch surfaces. In the Appendix, we will show how AS can be interpreted as categories naturally attached to conic bundles. Theorem3 could be considered as an analogue of Amitsur’s theorem that k-birational Severi– Brauer varieties have Brauer classes that generate the same subgroup of Br(k). In the case where d ≥ 5, we push this further to analyze how the algebras Ai obstruct the existence of closed points of given degree. The index ind(A) of a central simple k-algebra A is the greatest common divisor of the degrees of all central simple k-algebras Morita equivalent to A. We extend this definition to finite dimensional semisimple algebras by taking the least common multiple of the indices of simple components considered over their centers.

Theorem 4. Let S be a non-k-rational del Pezzo surface of degree d ≥ 5, and A1, A2 the associated semisimple algebras. If Ai has index di then ind(S) = d1d2/gcd(d1, d2). In particular: 2 • If d = 9, then d1 = d2 is either 1 if S ' Pk or 3 if S(k) 6= ∅. DEL PEZZO SURFACES 5

• If d = 8 and S is an involution surface, then (up to renumbering) A1 is a central simple 3 k-algebra, d1|4 and d2|2, and d1 = 1 if and only if S is a quadric in Pk while d2 = 1 if and only S is rational. • If d = 6, then (up to renumbering) d1|3 and d2|2, with S birationally rigid if di > 1. • In all other cases, d1 = d2 = 1 and S is rational. The key of the proof of Theorem4 is the explicit description of the vector bundles generating the exceptional blocks of S, together with an analysis of all possible Sarkisov links, cf. [58]. For del Pezzo surfaces of Picard rank one and degree at least 5, a consequence of our results is that the index of S can be calculated only in terms of the second Chern classes of generators of the 3 blocks, improving upon, in this case, a result of Hassett and Tschinkel [55, Lemma 8].

Theorem 5. Let S be a del Pezzo surface of degree d ≥ 5. There exist generators Vi for the three blocks such that ind(S) = gcd{c2(V0), c2(V1), c2(V2)}. 3 Moreover, unless d = 5, d = ind(S) = 6, or S is an anisotropic quadric surface in P then

ind(S) = gcd{c2(V1), c2(V2)} for indecomposable generators of the blocks of AS. Notice that more precise and detailed statements can be given in each specific case. Consult Sections6–10 and Tables2–5 for details. In summary, our results establish a complete understanding of the relationship between birational geometry, derived categories, and vector bundles on del Pezzo surfaces of degree ≥ 5. Structure. The paper is organized as follows. Part 1 organizes the necessary algebraic and geometric background. In Section1 we treat semiorthogonal decompositions and a generalized notion of exceptional objects, whose descent is treated in Section2. Section3 collects useful results on geometrically rational surfaces. Part 2 is dedicated to a detailed statement and proof of the main results. Before proceeding to the proofs, we recall the exceptional block theory in Section4. Section5 proves the main results for surfaces of low degree, while the higher degree cases are treated separately in Sections6–10. Part 3 consists of three appendices containing useful calculations related to elementary links. Acknowledgements. The authors are grateful to H. Esnault, whose initial questions formed one of the most important inspirations for this work. They would also like to acknowledge B. Antieau, J.-L. Colliot- Thélène,S. Gille, N. Karpenko, S. Lamy, R. Parimala, A. Quéguiner-Mathieu, T. Várilly-Alvarado, B. Viray, and Y. Tschinkel for useful conversations and remarks. This project was started the week following hurricane Sandy, during which time the Courant Institute at New York University graciously hosted the authors in one of the few buildings in lower Manhattan not completely without electricity. Yale University and the American Institute of Mathematics also provided stimulating working conditions for both authors to meet and are warmly acknowledged. The first author was partially sponsored by NSF grant DMS-0903039 and a Young Investigator grant from the NSA. Notations. Fix an arbitrary field k. If X is a k-scheme, we will denote by Db(X) the bounded derived category of complexes of coherent sheaves on X, considered as a k-linear category. If B is an OX -algebra, we will denote by Db(X,B) the bounded derived category of complexes of B-modules, considered as a k-linear category. If X = Spec(K) is an affine k-scheme and B is associated to a K-algebra, we will write Db(K/k, B) as shorthand. Also Db(K,A) is shorthand for Db(K/K, A). If A is the restriction of scalars of B down to k, we remark that Db(K/k, B) is k-equivalent to Db(k, A). Finally, if B is Azumaya with class β in Br(X), by abuse of notation, we will write Db(X, β) for Db(X,B). Triangulated categories will be commonly denoted with sans-serif font; dg-categories with calligraphic font; finite dimensional k-algebras with upper case Roman letters near the beginning of the alphabet; finite products of field extensions of k with l; vector bundles with upper case Roman letters; and Brauer classes as lower case Greek letters. 6 ASHER AUEL AND MARCELLO BERNARDARA

Contents Introduction 1

Part 1. Background on derived categories and geometrically rational surfaces 6 1. Semiorthogonal decompositions and categorical representability6 2. Descent for semiorthogonal decompositions 11 3. Geometrically rational surfaces 16

Part 2. Del Pezzo surfaces 19 4. Exceptional objects and blocks on del Pezzo surfaces 19 5. Del Pezzo surfaces of Picard rank 1 and low degree 21 6. Severi–Brauer surfaces (del Pezzo surfaces of degree 9) 25 7. Involution varieties (del Pezzo surfaces of degree 8) 26 8. del Pezzo surfaces of degree 7 29 9. del Pezzo surfaces of degree 6 29 10. del Pezzo surfaces of degree 5 36

Part 3. Appendix: Explicit calculations with elementary links 39 Appendix A. Elementary links for non-rational minimal del Pezzo surfaces 39 Appendix B. Links of type I, and minimal del Pezzo surfaces of “conic bundle type” 46 Appendix C. Links of type II between conic bundles of degree 8 48 References 50

Part 1. Background on derived categories and geometrically rational surfaces 1. Semiorthogonal decompositions and categorical representability We start by recalling the categorical notions that play the main role in this paper. Let k be an arbitrary ﬁeld. The theory of exceptional objects and semiorthogonal decompositions in the case where k is algebraically closed and of characteristic zero was studied in the Rudakov seminar at the end of the 80s, and developed by Rudakov, Gorodentsev, Bondal, Kapranov, and Orlov among others, see [52], [28], [29], [32], and [93]. As noted in [11], most fundamental properties persist over any base ﬁeld k.

1.1. Semiorthogonal decompositions and their mutations. Let T be a k-linear triangulated category. A full triangulated subcategory A of T is called admissible if the embedding functor admits a left and a right adjoint. Deﬁnition 1.1 ([29]). A semiorthogonal decomposition of T is a sequence of admissible subcategories A1,..., An of T such that

• HomT(Ai,Aj) = 0 for all i > j and any Ai in Ai and Aj in Aj; • for all objects Ai in Ai and Aj in Aj, and for every object T of T, there is a chain of morphisms 0 = Tn → Tn−1 → ... → T1 → T0 = T such that the cone of Tk → Tk−1 is an object of Ak for all k = 1, . . . , n. Such a decomposition will be written

T = hA1,..., Ani. If A ⊂ T is admissible, we have two semi-orthogonal decompositions T = hA⊥, Ai = hA,⊥ Ai, where A⊥ and ⊥A are, respectively, the left and right orthogonal of A in T (see [29, §3]). Given a semiorthogonal decomposition T = hA, Bi, Bondal [28, §3] deﬁnes left and right mutations LA(B) and RB(A) of this pair. In particular, there are equivalences LA(B) ' B and DEL PEZZO SURFACES 7

RB(A) ' A, and semiorthogonal decompositions

T = hLA(B), Ai, T = hB,RB(A)i. We refrain from giving an explicit deﬁnition for the mutation functors in general, which can be found in [28, §3]. In §1.2 we will give an explicit formula in the case where A and B are generated by exceptional objects.

1.2. Exceptional objects, blocks, and mutations. Very special examples of admissible subcategories, semiorthogonal decompositions, and their mutations are provided by the theory of exceptional objects and blocks. These constructions appear naturally on Fano varieties, and especially on geometrically rational surfaces, and they play a central role in our study. We provide here a detailed treatment, generalizing to any field notions usually studied over algebraically closed fields. Let T be a k-linear triangulated category. The triangulated category h{Vi}i∈I i generated by a class of objects {Vi}i∈I of T is the smallest thick (that is, closed under direct summands) full r triangulated subcategory of T containing the class. We will write ExtT(V,W ) = HomT(V,W [r]). Definition 1.2. Let A be a division (not necessarily central) k-algebra (e.g., A could be a field extension of k). An object V of T is called A-exceptional if r HomT(V,V ) = A and ExtT(V,V ) = 0 for r 6= 0. An exceptional object in the classical sense [51, Def. 3.2] of the term is a k-exceptional object. By exceptional object, we mean A-exceptional for some division k-algebra A. A totally ordered set {V1,...,Vn} of exceptional objects is called an exceptional collection if r ExtT(Vj,Vi) = 0 for all integers r whenever j > i. An exceptional collection is full if it generates T, equivalently, if for an object W of T, the vanishing HomT(W, Vi) = 0 for all i implies W = 0. r An exceptional collection is strong if ExtT(Vi,Vj) = 0 whenever r 6= 0. Remark 1.3. The extension of scalars of an exceptional object (in this generalized sense) need not be exceptional, see Remark 2.1 for more details. However, if k is algebraically closed, then all exceptional objects are automatically k-exceptional, hence remain exceptional under any extension of scalars. Exceptional collections provide examples of semiorthogonal decompositions when T is the bounded derived category of a smooth projective scheme.

Proposition 1.4 ([28, Thm. 3.2]). Let {V1,...,Vn} be an exceptional collection on the bounded derived category Db(X) of a smooth projective k-scheme X. Then there is a semiorthogonal decomposition b D (X) = hV1,...,Vn, Ai, where A is the full subcategory of objects W such that HomT(W, Vi) = 0. In particular, the sequence if full if and only if A = 0.

Given an exceptional pair {V1,V2} with Vi being Ai-exceptional, consider the admissible subcategories hVii, forming a semiorthogonal pair. We can hence perform right and left mutations, which provide equivalent admissible subcategories. Recall that mutations provide equivalent admissible subcategories and ﬂip the semiorthogo- nality condition. It easily follows that the object RV2 (V1) is A1-exceptional, the object LV1 (V2) is

A2-exceptional, and the pairs {LV1 (V2),V1} and {V2,RV2 (V1)} are exceptional. We call RV2 (V1) the right mutation of V1 through V2 and LV1 (V2) the left mutation of V2 through V1. In the case of k-exceptional objects, mutations can be explicitly computed.

Definition 1.5 ([51, §3.4]). Given a k-exceptional pair {V1,V2} in T, the left mutation of V2 with respect to V1 is the object LV1 (V2) defined by the distinguished triangle: ev (1.1) HomT(V1,V2) ⊗ V1 −−→ V2 −→ LV1 (V2), 8 ASHER AUEL AND MARCELLO BERNARDARA where ev is the canonical evaluation morphism. The right mutation of V1 with respect to V2 is the object RV2 (V1) defined by the distinguished triangle: coev RV2 (V1) −→ V1 −−→ HomT(V1,V2) ⊗ V2, where coev is the canonical coevaluation morphism.

Given an exceptional collection {V1,...,Vn}, one can consider any exceptional pair {Vi,Vi+1} and perform either right or left mutation to get a new exceptional collection. Exceptional collections provide an algebraic description of admissible subcategories of T. In- deed, if V is an A-exceptional object in T, the triangulated subcategory hV i ⊂ T is equivalent to Db(k, A). The equivalence Db(k, A) → hV i is obtained by sending the complex A concen- trated in degree 0 to V . Moreover, as shown by Bondal [28] and Bondal–Kapranov [30], full exceptional collections give an algebraic description of a triangulated category. Proposition 1.6 ([28, Thm. 6.2]). Suppose that T is the bounded derived category of either a smooth projective k-scheme or a k-linear abelian category with enough injective objects. Let Ln {V1,...,Vn} be a full strong exceptional collection on T, and consider the object V = i=1 Vi b and the k-algebra A = EndT(V ). Then RHomT(V, −): T → D (k, A) is a k-linear equivalence. Remark 1.7. The assumption on the category T and on the strongness of the exceptional sequence may seem rather restrictive, and both find a natural solution when triangulated categories are enriched with a dg-category structure. The first assumption can be indeed replaced by considering a dg-enhancement of T (see [30, Thm. 1]). When the exceptional collection is not strong, dg-algebras are required. See §1.3 for details. b n Example 1.8. The full strong k-exceptional collection {O, O(1),..., O(n)} on D (Pk ) is due to Ln Beilinson [17], [18] and Bernˇste˘ın–Gelfand–Gelfand[24]. In this case A = End i=0 O(i) is isomorphic to the path algebra of the Beilinson quiver with n + 1 vertices, see [28, Ex. 6.4]. We remark that the semisimplification of A is the algebra kn+1. Proposition 1.6 provides an alternative approach to exceptional collections and semiorthogonal decompositions by considering tilting objects. In this paper, we will not use this approach, but we find the language convenient at times, especially in relation to issues of Galois descent.

Deﬁnition 1.9. Given a strong exceptional collection {V1,...,Vm} of a k-linear triangulated Lm category T, the object V = i=1 Vi is called a tilting object for the subcategory hV1,...,Vmi. b Recall that Proposition 1.6 implies that hV1,...,Vmi ' D (k, EndT(V )). If the Vi are vector bundles on a smooth projective variety X, we call V a tilting bundle. The existence of a tilting bundle also yields, analogously to Proposition 1.6, a presentation in algebraic K-theory.

Proposition 1.10. Let {V1,...,Vn} be a full strong exceptional collection of vector bundles on Ln a smooth projective k-variety X and let V = i=1 Vi be the associated tilting bundle. Suppose that Vi is Ai-exceptional. Then HomX (V, −): Ki(X) → Ki(A1 × · · · × An) is an isomorphism for all i ≥ 0.

Proof. Letting A = EndDb(X)(V ). Since the exceptional objects are vector bundles, HomX (V, −) deﬁnes a morphism from the category of vector bundles on X to the category of A-modules. b b Proposition 1.6 implies that RHomDb(X)(V, −): D (X) → D (k, A) is an equivalence. Hence we can apply [107, Thm. 1.9.8, §3], implying that HomX (V, −): Ki(X) → Ki(A) is an isomorphism ∼ for all i ≥ 0. But A has a nilpotent ideal I so that A/I = A1 × · · · × An, so the result follows ∼ from the fact that Ki(A) = Ki(A/I). Remark 1.11. One can prove the same result for any additive invariant of Db(X) via noncommutative motives and Tabuada’s universal additive functor (see [101]).

A special case of an exceptional pair is a completely orthogonal pair {V1,V2}, i.e., an exceptional pair such that {V2,V1} is also exceptional. In this case, RV2 (V1) = V1 and LV1 (V2) = V2. DEL PEZZO SURFACES 9

Deﬁnition 1.12 ([64], 1.5). An exceptional block in a k-linear triangulated category T is an r exceptional collection {V1,...,Vn} such that ExtT(Vi,Vj) = 0 for every r whenever i 6= j. Equivalently, every pair of objects in the collection is completely orthogonal. By abuse of notation, we denote by E the exceptional block as well as the subcategory that it generates. Ln If E is an exceptional block, then EndT( i=1 Vi) is isomorphic to the k-algebra A1 ×· · ·×An, b where Vi is Ai-exceptional. Proposition 1.6 then yields a k-equivalence E ' D (A1 × · · · × An). Moreover, given an exceptional block, any internal mutation acts by simply permuting the exceptional objects. Given an exceptional collection {V1,...,Vn,W1,...,Wm} consisting of two blocks E and F, the left mutation LE(F) and the right mutation RF(E) are obtained by mutating all the objects of one block to the other side of all the objects of the other block, or, equivalently, as mutations of semiorthogonal admissible subcategories.

1.3. dg-enhancements. Recall that a k-linear dg-category A is a category enriched over dg- complexes of k-vector spaces, that is, for any pair of objects x, y in A , the morphism set HomA (x, y) has a functorial structure of a diﬀerential graded complex of vector spaces. The 0 0 category H (A ) has the same objects as A , and HomH0(A )(x, y) = H (HomA (x, y)), in particular H0(A ) is triangulated. We will only consider pretriangulated dg-categories (see [66, §4.5]). A semiorthogonal decomposition of A = hB1,..., Bni is a set of pretriangulated full 0 0 0 subcategories such that hH (B1),...,H (Bn)i is a semiorthogonal decomposition of H (A ). Consult [66] for an introduction to dg-categories. Let X and Y be a smooth proper k-schemes, and recall that a Fourier–Mukai functor of b b b kernel P ∈ D (X × Y ) is the exact functor Φ : D (X) → D (Y ) given by the formula ΦP (−) = ∗ L (RpY )∗(pX (−) ⊗ P ) (see, e.g., [56, §5]). Considering the bounded derived category Db(X) as a triangulated k-linear category gives rise to some functorial problems. One of them has a geometric nature: given an exact functor Φ: Db(X) → Db(Y ) it is not known whether it is isomorphic to a Fourier–Mukai functor, except in some special cases, see [37]. For example, suppose that AX and AY are admissible triangulated categories of Db(X) and Db(Y ), respectively, and suppose that there is a triangulated equivalence AX ' AY . The composition functor

b b Φ: D (X) AX ' AY ,→ D (Y ) is conjectured to be a Fourier–Mukai functor ([73, Conj. 3.7]). As proved by Lunts and Orlov [80], there is a unique functorial enhancement Db(X) for Db(X), that is a dg-category such that H0(Db(X)) = Db(X). Given X and Y as before, a functor Φ : Db(X) → Db(Y ) is of Fourier–Mukai type if and only if there is a functor Φ : Db(X) → Db(Y ) such that H0(Φ) = Φ, see [23, Prop. 9.4] for example, or [81] for a more general statement. On the other hand, Kuznetsov [74] has shown that, for any admissible subcategory AX ⊂ Db(X), the projection functor (i.e., the right adjoint to the embedding functor) is a Fourier– Mukai functor. Hence, the choice of such a functor endows AX with a dg-structure, which is in principle not unique, but depends on the choice of projection. Hence, when dealing with a semiorthogonal decomposition b D (X) = hA1,..., Ani, one always has a decomposition

b D (X) = hA1,..., Ani,

b where the dg-category D (X) is unique, though the dg-enhancements Ai of Ai may not be. Finally, we recall that if there exists a smooth projective scheme Z and a Brauer class α b b b in Br(Z) such that Ai ' D (Z, α), then the embedding functor Φ : D (Z, α) → D (X) is a Fourier–Mukai functor [36], and hence the dg-structure Db(Z, α) is unique. Other speciﬁc cases are provided by Homological Projective Duality [73]. 10 ASHER AUEL AND MARCELLO BERNARDARA

1.4. Categorical representability. Using semiorthogonal decompositions, one can deﬁne a notion of categorical representability for a triangulated category. In the case of smooth projective varieties, this is inspired by the classical notions of representability of cycles, see [21]. Deﬁnition 1.13 ([21]). A k-linear triangulated category T is representable in dimension m if it admits a semiorthogonal decomposition

T = hA1,..., Ari, and for each i = 1, . . . , r there exists a smooth projective connected k-variety Yi with dim Yi ≤ b m, such that Ai is equivalent to an admissible subcategory of D (Yi). Deﬁnition 1.14 ([21]). Let X be a projective k-variety of dimension n. We say that X is categorically representable in dimension m (or equivalently in codimension n−m) if there exists a categorical resolution of singularities of Db(X) that is representable in dimension m. The following explains why categorical representability in codimension 2 is conjecturally related to birational geometry in the spirit of Question1. Lemma 1.15. Let X → Y be the blow-up of a smooth projective k-variety along a smooth center. If Y is categorically representable in codimension 2 then so is X. Proof. Denoting by Z ⊂ Y the center of the blow-up f : X → Y , which has codimension at least 2 in Y , Orlov’s blow-up formula [84] gives a semiorthogonal decomposition Db(X) = hf ∗Db(Y ), Db(Z)i. As X and Y have the same dimension, if Y is categorically representable in codimension 2, then so is X.

An additive category T is indecomposable if for any product decomposition T ' T1 × T2 into additive categories, we have that T ' T1 or T ' T2. Equivalently, T has no nontrivial completely orthogonal decomposition. Remark that if X is a k-scheme then Db(X) is indecomposable if and only if X is connected (see [33, Ex. 3.2]). More is known if X is the spectrum of a field or a product of fields. Lemma 1.16. Let K be a k-algebra. (i) If K is a field and A is a nonzero admissible k-linear triangulated subcategory of Db(k, K), then A = Db(k, K). ∼ (ii) If K = K1 ×· · ·×Kn is a product of field extensions of k and A is a nonzero admissible b b indecomposable k-linear triangulated subcategory of D (k, K), then A ' D (k, Ki) for some i = 1, . . . , n. ∼ (iii) If K = K1 ×· · ·×Kn is a product of field extensions of k and A is a nonzero admissible b Q b k-linear triangulated subcategory of D (k, K), then A ' j∈I D (k, Kj) for some subset I ⊂ {1, . . . , n}. Proof. For (i), it suffices to note that any nonzero object of Db(k, K) is a generator. For (ii), b b b with respect to the product decomposition D (k, K) ' D (k, K1) × · · · × D (k, Kn), consider b the projections xi to D (k, Ki) of an object x of A. Any nonzero xi will generate the respective b subcategory D (k, Ki). Given a nonzero object x in A, there exist 1 ≤ i ≤ n such that xi 6= 0. If there exists another object y in A with yj 6= 0 for j 6= i, then the objects xi and yj will generate completely orthogonal nontrivial subcategories of A. Since A is indecomposable, this is impossible. Hence b for every object x in A, we have that xj = 0 for every j 6= i. In particular, A ⊂ D (k, Ki), hence they are equal by (i). For (iii), we apply (ii) to the indecomposable components of A. Next, we need the following affine case of a conjecture of Cˇaldˇararu[35, Conj. 4.1]. The simple proof below is taken from the unpublished manuscript [13] as part of a proof of Cˇaldˇararu’s conjecture in the relative case. A proof of the conjecture (using different techniques) was recently obtained by Antieau [5], with the case of arbitrary (possibly non-torsion) classes over general spaces handled by [34], after progress by [36] and [90]. Recall that a central simple k-algebra A is a simple k-algebra whose center is k. More generally, if X is a scheme, an Azumaya algebra A over X is a locally free OX -algebra of DEL PEZZO SURFACES 11

op ﬁnite rank such that A ⊗R A is isomorphic to the matrix algebra End(A). Azumaya algebras A and B are Brauer equivalent if there exist locally free OX -modules P and Q such that A ⊗ End(P ) =∼ B ⊗ End(Q), and they are Morita equivalent over X if their stacks of coherent modules over X are equivalent. Then Brauer equivalence coincides with Morita equivalence over X, and the group of equivalence classes under tensor product is the Brauer group Br(X). For an Azumaya algebra A over X, we write An for A⊗n for n ≥ 0 and (Aop)⊗−n for n ≤ 0. Theorem 1.17. Let R be a noetherian commutative ring, U and V be R-algebras, and A and B be Azumaya algebras over U and V , respectively. Then A and B are Morita R-equivalent if and only if there exists an R-algebra isomorphism σ : U → V such that A and σ∗B are Brauer equivalent over U. Proof. First suppose that A and σ∗B are Brauer equivalent, where σ : U → V is an R-algebra isomorphism. Then by Morita theory for Azumaya algebras (cf. [65, §19.5]), there is a Morita U- equivalence Coh(U, A) → Coh(U, σ∗B), which by restriction of scalars, is a Morita R-equivalence. Also, (σ−1)∗ : Coh(U, σ∗B) → Coh(V,B) is an R-equivalence. Composing these yields the required Morita R-equivalence. Now suppose that F : Coh(U, A) → Coh(V,B) is a Morita R-equivalence. Since F is essen- tially surjective, we can choose P ∈ Coh(U, A) such that F (P ) =∼ Bop as B-modules. Since F is fully faithful, any choice of isomorphism θ : F (P ) =∼ Bop deﬁnes a left B-module structure on P . In this way, P has a B-A-module structure with commuting R-structure. In particular, for 0 0 any A-module P , HomA(P,P ) has a natural right B-module structure via precomposition. The equivalence F being R-linear yields an induced R-algebra isomorphism ∼ ∼ op ψ : EndA(P ) = HomA(P,P ) = HomB(F (P ),F (P )) = EndB(B ) = B. In particular, ψ restricts to an R-algebra isomorphism σ : U → V of the centers, hence becomes a ∗ ∗ op ∗ U-module isomorphism EndA(P ) → σ B = (σ B) , so that A is Brauer equivalent to σ B. Corollary 1.18. Let A and B be simple k-algebras with respective centers K and L. Then Db(K/k, A) and Db(L/k, B) are (k-linearly) equivalent if and only if there exists some k- automorphism σ : K → L such that A and σ∗B are Brauer equivalent over K. Proof. By [110, Cor. 2.7], if A or B is either commutative or local artinian (not necessarily commutative), then they are derived k-equivalent if and only if they are Morita k-equivalence. Then we apply Theorem 1.17. We remark that Corollary 1.18 is also a consequence of the results of [6]. Lemma 1.19. A k-linear triangulated category T is representable in dimension 0 if and only if there exists a semiorthogonal decomposition

T = hA1,..., Ari, b such that for each i, there is a k-linear equivalence Ai ' D (Ki/k) for an étale k-algebra Ki. Proof. The smooth k-varieties of dimension 0 are precisely the spectra of étale k-algebras. Hence the semiorthogonal decomposition condition is certainly sufficient for the representability of T in dimension 0. On the other hand, if T is representable in dimension 0, we have that each Ai an admissible subcategory of the derived category of an étale k-algebra. By Lemma 1.16(iii), we have that Ai is thus itself such a category. 2. Descent for semiorthogonal decompositions Given a smooth projective variety X over a field k, and fixing a separable closure ks of k, we are interested in comparing k-linear semiorthogonal decompositions of Db(X) and ks -linear b semiorthogonal decompositions of D (Xks ). The general question of how derived categories and semiorthogonal decompositions behave under base field extension has started to be addressed by several authors [3, §2], [4], [7], [47], [74], [97], [105]. Galois descent does not generally hold for objects in a triangulated category, due to the fact that cones are only defined up to quasi-isomorphism. However, for a linearly reductive algebraic group G acting on a variety X, 12 ASHER AUEL AND MARCELLO BERNARDARA a general theory of descent of semiorthogonal decompositions has been developed by Elagin [47, 46, 45]. In the case where K/k is a finite G-Galois extension and we consider G acting on XK via k-automorphisms, this theory works as long as the characteristic of k does not divide the order of G. We are then faced with two main questions. First, given a ks -linear triangulated category T, what are all the k-linear triangulated categories T such that Tks is equivalent to T? This first question is very challenging (see [4] for some examples), and we usually restrict ourselves to consider Tks in a restricted class of categories (e.g., those generated by a strong exceptional collection). Notice that we are actually dealing with triangulated categories which admit a canonical dg-enhancement, and this should be the natural structure to consider. Second, given a set of admissible subcategories in Db(X), determine how this can be related to the b semiorthogonal decomposition of D (Xks ). In our geometric applications, a central rôleis played by descent of exceptional blocks and vector bundles.

2.1. Scalar extension of triangulated categories. Let X be a k-scheme and V an OX - module. For a ring extension K/k we write XK = X ×Spec k Spec K and fK : XK → X for the ∗ projection and VK = fK V for the extension of V to K. If K is a ﬁnite k-algebra and F is an

OXK -module, we denote by trK/kF = (fK )∗F the trace of F from K down to k. We similarly b define the trace of an object of D (XK ). If F is locally free of rank r on XK then trK/kF is s locally free of rank r[K : k] on X. We denote by Gk = Gal(k /k) the absolute Galois group of k. If T is a k-linear triangulated category with a dg-enhancement and K/k is a field extension, then denote by TK the K-linear extension of scalars category defined in [97]. This category is a thick dg-enhanced K-linear triangulated category. As expected, there is a K-linear equivalence b b D (X)K ' D (XK ). If K/k is a Galois extension, then the Galois group acts by k-linear equivalences on TK . If K/k is a finite G-Galois extension then G acts on XK as a k-scheme b b and there is a k-linear equivalence D (X) ' DG(XK ) with the bounded derived category of G-equivariant coherent sheaves on XK considered as a k-scheme. Given an orthogonal pair of admissible subcategories {A, B} in Db(X), we can perform right and left mutations. These operations commute with base change, i.e., LA(B)K = LAK (BK ), for any finite extension K of k, and similarly for the right mutation. Remark 2.1. Assume that [K : k] is finite and that X is a smooth projective k-variety. For b b b an object V in D (X), the scalar extension VK in D (X)K = D (XK ) satisfies EndTK (VK ) = EndT(V ) ⊗k K, see [53, §0.5.4.9, 1.9.3.3]. If A is a division k-algebra, then A ⊗k K may not be. b Hence if V is an exceptional object in D (X), then VK may fail to be an exceptional object in b D (XK ). However, if V is k-exceptional, then VK is always K-exceptional. Lemma 2.2. Let T be an admissible k-linear subcategory of Db(X) for a smooth projective b k-variety X. Let K be any extension of k. Then TK is admissible in D (XK ). Proof. By Kuznetsov [74], the inclusion functor of the admissible subcategory T → Db(X) has an adjoint ρ : Db(X) → T that is of Fourier–Mukai type with kernel P an object in Db(X × X). b Taking P ⊗k K in D (XK × XK ) as the kernel of a Fourier–Mukai functor, we obtain an adjoint b for TK → D (XK ). Based on base change results for Fourier–Mukai functor due to Orlov (see [85]), the following statement was proved in [11, Lemma 2.9] (see also [3, Prop. 2.1]). Lemma 2.3. Let X be a smooth projective variety over k, and K a finite extension of k. Suppose b b that T1,..., Tn are admissible subcategories of D (X) such that D (XK ) = hT1K ,..., TnK i. b Then D (X) = hT1,..., Tni. 2.2. Classical descent theory for vector bundles. For a vector bundle V on a scheme X, denote by A(V ) = End(V )/rad(End(V )) and Z(V ) the center of A(V ). Assuming that X is a proper k-scheme, then vector bundles on X enjoy a Krull–Schmidt decomposition, hence A(V ) DEL PEZZO SURFACES 13

Lm ⊕di is a semisimple k-algebra. If V = i=1 Vi is the Krull–Schmidt decomposition of V , then

A(V ) is the product of the algebras Mdi (A(Vi)). In particular, V is indecomposable on X if and only if A(V ) is a division algebra over k. If K/k is a ﬁeld extension and E is a vector bundle, then we have End(VK ) = End(V )K (cf. Remark 2.1), and hence A(VK ) = A(V )K if K/k is separable. Lemma 2.4 ([9, Lemma 1.1]). Let V be an indecomposable vector bundle on X. Let K/k be a normal extension containing Z(V ) and splitting A(V ). Write m = [Z(V ): k]sep and Lm ⊕d d = degZ(V )(A(V )). Then VK has a Krull–Schmidt decomposition of the form VK = i=1 Vi where Vi are indecomposable over XK and A(Vi) = K.

Let W be an indecomposable vector bundle on Xks . A vector bundle V on X is pure (of type W ) if Vks is a direct sum of vector bundles isomorphic to W .

Proposition 2.5 ([9, Prop. 3.4]). Let X be a proper variety over k. If W be a Gk-invariant indecomposable vector bundle on Xks then there exists a pure indecomposable vector bundle V on X, unique up to isomorphism, of type W .

Deﬁnition 2.6. Given a Gk-invariant indecomposable vector bundle W on Xks , let V be the vector bundle on X given in Proposition 2.5. Deﬁne α(W ) ∈ Br(k) to be the Brauer class of A(V ). In fact, α(W ) is the Brauer class of A(V 0) for any pure vector bundle V 0 on X of type W . Remark 2.7. Recall the Brauer obstruction for Galois invariant line bundles on a smooth proper geometrically integral variety X over k, see [100, §2]. The sequence of low degree terms of the Leray spectral sequence for Gm associated to the structural morphism X → Spec k is

Gk d (2.1) 0 → Pic(X) → Pic(Xks ) −→ Br(k) → Br(X). The differential d is called the Brauer obstruction for a Galois invariant line bundle; a class in G Pic(Xks ) k descends to a line bundle on X if and only if it has trivial Brauer obstruction. The G cokernel of Pic(X) → Pic(Xks ) k is torsion since the Brauer group is. We also recall that if A is a central simple algebra with class d(L), for a Galois invariant line bundle L, and Y is the Severi–Brauer variety associated to A, then up to tensoring by elements of Pic(X), we can assume that L corresponds to a morphism X → Y , see [100, Prop. 8]. Corollary 2.8 ([9, Prop. 3.15]). Let r be the index of the Brauer class α(W ) ∈ Br(k). Then ∼ ⊕r Gk Vks = W , and hence c1(Vks ) = rc1(W ) is in the image of Pic(X) → Pic(Xks ) . In particular, W descends to X if and only if α(W ) is trivial Theorem 2.9 ([9, Thm. 1.8]). Let X be a proper variety over k and V an indecomposable vector bundle on X. If K is a maximal subfield of A(V ) then there is an absolutely indecomposable ∼ vector bundle W on XK such that V = trK/kW . Definition 2.10. Let V be a vector bundle on a proper smooth scheme X over k. We write n1 nr V = V1 ⊕ · · · ⊕ Vr the Krull–Schmidt decomposition, with V1,...,Vr indecomposable. The min min reduced part of V is defined to be V = V1 ⊕ · · · ⊕ Vr. We remark that V and V generate the same thick subcategory of Db(X).

2.3. Descent of exceptional blocks. If Tks admits a full exceptional block, we wish to classify all the k-linear categories T base-changing to Tks . In particular, if Tks is generated by an b exceptional block of vector bundles (i.e., it admits a tilting bundle) in D (Xks ) for some smooth b proper scheme X, the descent of Tks to T inside D (X) is described by descent of the tilting bundle. s When Tks is generated by a k -exceptional object, the descent question has been studied by To¨en[105]. Recall (see Corollary 1.18) that Brauer equivalence between Azumaya algebras over a ring R is the same as Morita R-equivalence, and that Morita k-equivalence between local (possibly noncommutative) k-algebras is the same as derived k-equivalence by [110, Cor. 2.7]. 14 ASHER AUEL AND MARCELLO BERNARDARA

Theorem 2.11 (Toën[105, Cor. 2.15]). If T is a k-linear triangulated category such that b s b Tks = D (k ), then there exists a central simple algebra A over k such that T = D (k, A). In particular, T is generated by a k-exceptional object if and only if A is trivial in the Brauer group. Proof. Since Db(ks ) has compact generator, then so does T by [105, Prop. 4.12]. Then the dg- b category T is equivalent to D (k, A) for some dg-algebra A over k. As Aks is Morita equivalent to ks and ks /k is faithfully flat, then by [105, Prop. 2.3], A is a derived Azumaya algebra over k. By [105, Prop. 2.12], A is Morita equivalent to an Azumaya algebra over k. s ∼ s s Recall that K is a finite étale k-algebra if and only if K ⊗k k = k × · · · × k . Equivalently, ∼ K = l1 × · · · × lm where each li/k is a finite separable field extension. The k-dimension of K ∼ is called the degree of K over k. An Azumaya algebra A over K = l1 × · · · × lm is simply a ∼ product A = A1 × · · · × Am where each Ai is a central simple li-algebra. A separable algebra of dimension nr2 over k is an Azumaya algebra of degree r over an étalealgebra K over k of dimension n. The set of k-isomorphism classes of étale k-algebras of degree n is in bijection with the Galois 1 cohomology set H (k, Sn). The set of k-isomorphism classes of degree n Azumaya algebras over an étale k-algebra of degree r is in bijection with the Galois cohomology set H1(k, G), where G n is the group scheme defined by the functor G(R) = AutR(Mr(R )). Note that G is isomorphic n to the wreath product PGLr o Sn = PGLr o Sn with Sn acting via permutation of the factors. 1 1 Then there is a natural map H (k, G) → H (k, Sn) given by taking the isomorphism class of the center (this map is clearly surjective), whose fiber over the class of an étalealgebra K of dimension n over k is given by the set of isomorphism classes of Azumaya algebras over K of degree r. Finally, by the classical theory of central simple algebras over a field, each central simple k-algebra A of degree n contains a maximal étale k-subalgebra K of degree n splitting A, i.e., ∼ such that A ⊗k K = Mn(K). s Proposition 2.12. Let T be a k-linear triangulated category such that Tks is k -equivalent to Db(ks , (ks )n). Then there exists an étalealgebra K of degree n over k, an Azumaya algebra A over K, and a k-linear equivalence T ' Db(K/k, A). In this case, T is an indecomposable category if and only if K is a field extension of k.

Proof. Toën[105, Cor. 3.12] shows that the derived group stack of autoequivalences AutdAz/k(k) of the trivial derived Azumaya k-algebra k is equivalent to Z×K(Gm, 1), as any dg-autoequivalence of an étale k-algebra R is given by tensoring with some invertible perfect dg-R-module, which by [106, Thm. 8.15], are all of the form L[n] for an invertible R-module L and n ∈ Z. n As a generalization, we claim that the derived group stack of autoequivalences Autdg/k(k ) of the étale k-algebra kn (thought of as a dg-algebra over k), is equivalent to the wreath product (Z × K(Gm, 1)) o Sn, thought of as n × n generalized permutation matrices filled with shifts of invertible modules. Indeed, for an étale k-algebra R, any dg-endofunctor of Rn can be viewed as an n × n matrix M of perfect dg-R-modules by [106, Thm. 8.15]. Considering the n × n matrix M of ranks of the entries of M, we see that M is invertible implies that M is invertible and its inverse also consists of nonnegative integers, which is well-known to imply that M is a permutation matrix (i.e., GLn(N) = Sn). Hence M must have exactly a single nonzero module in each row and column, and this entry must have rank 1, giving the desired form. n We are now interested in the stack F = Formsdg/k(k ) associated to the prestack of dg- n n algebras étalelocally Morita equivalent to k , which is simply K(Autdg/k(k ), 1), as in [105, Cor. 3.12]. The exact sequence of group stacks n n 1 → (Z × K(Gm, 1)) → Autdg/k(k ) → Sn → 1 gives rise to a fibration of stacks n (K(Z, 1) × K(Gm, 2)) → F → K(Sn, 1). Hence to any dg-algebra A étalelocally Morita equivalent to kn, we have a class φ(A) in F(k) (the sections over k of the stack F) and the association A 7→ φ(A) is injective. Hence F(k) fits DEL PEZZO SURFACES 15 into an exact sequence n 1 Br(k) → F(k) → H (k, Sn) → 1, 1 1 (using that H (k, Z) = 0) where the map z : F(k) → H (k, Sn) sends φ(A) to the isomorphism class of the center Z(A). By a standard cohomological twisting argument, the fiber of z over an étalealgebra K of degree n over k, is the image of Br(K). This gives the desired description. The final claim follows since for any field extension K over k, the k-linear category Db(K/k, A) is indecomposable; conversely, given any decomposition K = K1 × K2, there is an induced b b decomposition A = A1 ×A2, with Ai Azumaya over Ki, and then D (K/k, A) ' D (K1/k, A1)× b D (K2/k, A2). Let us state explicitly the following corollary which will be extensively used in our applications. Corollary 2.13. Let X be a smooth projective k-variety, K/k a Galois extension with group b G. Suppose that there is an admissible subcategory T of D (X) such that TK = hV1,...,Vni is an exceptional block of vector bundles on XK . Then for any object A in T the Chern class c1(A) is a Z-linear combination of the c1(Vi). In particular, there exists a nonzero G-invariant Z-linear combination of the c1(Vi). Proof. Proposition 2.12 implies that T has cohomological dimension 0, so that every object is the direct sum of its cohomologies, and this decomposition is clearly G-invariant. By the thickness of T, we can then suppose that A is a sheaf. Then AK is a direct sum of the vector bundles Vi, and is nontrivial if A 6= 0. Let us end this section by illustrating Lemma 2.3 by examples known in the literature of descent of vector bundles. Example 2.14. Let X be the Severi–Brauer variety associated to a central simple algebra A of degree n + 1 over k, see Example 3.2 for definitions. Let K/k be a field extension such that n XK ' PK . Thanks to Beilinson [17], we have the following semiorthogonal decomposition b n (2.2) D ( ) = h n , n (1),..., n (n)i PK OPK OPK OPK On the other hand, Db(X) admits the following semiorthogonal decomposition1 [19, Cor. 4.7] (2.3) Db(X) = hDb(k), Db(k, A),..., Db(k, An)i. We can see the semiorthogonal decomposition (2.2) as the base change of (2.3) using descent of vector bundles as follows. The exceptional line bundle n clearly descends to X, and hence OPK b b generates D (k) inside D (X). Consider now the exceptional line bundle n (1), and consider OPK ⊕n+1 it as a stand-alone block. It is well known (cf. [91, §8.4]), that V = n (1) descends to X OPK and that End(V ) = A. Similar arguments apply to n (i). Hence the decomposition (2.3) of OPK b b n D (X) can be obtained by descending the exceptional collection (2.2) of D (PK ). Other known (similar) examples include: the decomposition of a relative Severi–Brauer variety [19], whose base change is the decomposition of a projective bundle given by Orlov [84]; the decomposition of a generalized Severi–Brauer variety given by Blunk [26], whose base change is the decomposition of a Grassmannian variety given by Kapranov [60]; and the decomposition of a quadric hypersurface given in [11] (generalizing the work of Kuznetsov [71]), whose base change is the decomposition of a quadric given by Kapranov [60]. Remark 2.15. One might wonder, motivated by the previous examples, whether the descent of b an exceptional block E ⊂ D (Xks ), generated by vector bundles Vi can be realized by the descent L of the tilting bundle V = i Vi. If V were Galois invariant, the results of §2.2 would produce a vector bundle W on X pure of type V . Moreover, this would explicitly let us consider the

1In [19], the full and faithful embedding of the subcategory of A−1-modules into Db(X) is given by the choice of an “´etalelocal form” F of OX (1); an A-module M is sent to M ⊗F , which can be endowed with the structure −1 of an OX -module. Notice that in loc.cit. A is the algebra obstructing the descent of O(1), so that our notations are opposite to the ones used there. 16 ASHER AUEL AND MARCELLO BERNARDARA endomorphism algebra End(W ) to obtain an algebraic description of the descended category. We wonder whether a converse statement holds: if the block E descends, then is the tilting bundle V Galois-invariant?

3. Geometrically rational surfaces In this section we collect together some known results on geometrically rational surfaces, including their classiﬁcation, Chow groups, and derived categories.

3.1. Classification and first properties. Let k be a field, ks a separably closure, and k¯ an algebraic closure. A smooth projective geometrically integral surface S over k such that ¯ ¯ S = S ×k k is k-rational is called a geometrically rational surface. Recall that S is a del Pezzo ∨ surface if ωS is ample. The degree of a geometrically rational surface is the self-intersection number d = ωS · ωS. 2 We say that a field extension l of k is a splitting field for S if S ×k l is birational to Pl via a sequence of monoidal transformations centered at closed l-points. An important fact is that geometrically rational surfaces are separably split. The following result allows one to consider the separable closure ks instead of the algebraic closure. Proposition 3.1 ([42, Thm. 1], [108, Thm. 1.6]). If S is a geometrically rational surface over a field k, then S is split over ks . A surface S is minimal over k, or k-minimal, if any birational morphism f : S → S0, defined over k, is an isomorphism. Over a separably closed field, the only minimal rational surfaces are the projective plane and projective bundles over the projective line. Over a general field, this is no longer true. Minimal geometrically rational surfaces have been completely classified, and we have the following list (see [82], and the recent [54]): 2 (i) S = Pk is a projective plane, so Pic(S) = Z, generated by the hyperplane O(1); 3 (ii) S ⊂ Pk is a smooth quadric and Pic(S) = Z, generated by the hyperplane section O(1); (iii) S is a del Pezzo surface with Pic(S) = Z, generated by the canonical class ωS; (iv) S is a conic bundle f : S → C over a geometrically rational curve, with Pic(S) ' Z⊕Z. Example 3.2. A Severi–Brauer variety is a variety X over k such that X ∼ n−1 for some = Pk¯ n ≥ 2. The set of isomorphism classes of Severi–Brauer varieties X of dimension n − 1 over k is in bijection with the set of k-isomorphism classes of central simple algebras A of degree n over k, and we will write X = SB(A) accordingly. By a theorem of Châtelet,a Severi–Brauer variety X is k-rational if and only if X is k-isomorphic to projective space if and only if X(k) 6= ∅, cf. [50, Thm. 5.1.3]. In this case, we say that X splits and remark that X always splits after a n−1 finite separable field extension. The Galois action on Pic(Xks ) = Pic(Pks ) = Z is trivial since it preserves dimensions of spaces of global sections. It is a theorem of Lichtenbaum that the Brauer obstruction (see Remark 2.7) for the Galois invariant class of O 2 (1) to descend to X is Pks precisely the Brauer class of A, cf. [50, Thm. 5.4.10]. Thus for a nonsplit Severi–Brauer variety X, the low terms of the Leray spectral sequence (see (2.1)) show that ωX generates Pic(X). A Severi–Brauer surface S is a Severi–Brauer variety of dimension 2, hence is a minimal del Pezzo surface. As intersection numbers do not change under scalar extension, S has degree 9. By the above analysis of the Picard group, a nonsplit Severi–Brauer surface belongs to the case 2 (iii), while the split Severi–Brauer surface Pk belongs to case (i). Example 3.3. An involution variety is a variety X over k such that X is k¯-isomorphic to a n smooth quadric in Pk¯ for some n ≥ 2. The set of isomorphism classes of involution varieties over k is in bijection with the set of k-isomorphism classes of central simple algebras (A, σ) of degree n + 1 over k together with a quadratic pair σ, a generalization to arbitrary characteristic of the notion of an orthogonal involution, see [67, §5.B] for definitions. In this case, X is a degree 2 hypersurface in the Severi–Brauer variety SB(A). Attached to an involution variety X is the even Clifford algebra C0(A, σ), defined by Jacobson [59] via Galois descent in characteristic not 2 and by [67, §8] in general. If A is split, then the quadratic pair on A is adjoint to a DEL PEZZO SURFACES 17

∼ n quadratic form, uniquely defined up to scaling and X ⊂ SB(A) = Pk is the associated quadric hypersurface. We say that X is an anisotropic quadric if A is split yet X(k) = ∅. From now on, we assume that X has even dimension. In this case, C0(A, σ) is an Azumaya algebra over its center l, which is an étalequadratic extension of k, called the discriminant ∼ 2 extension of X. The Galois action on Pic(Xks ) = Z , since it preserves dimensions of spaces of global sections factors through a permutation of the divisors defining the two rulings of the quadric over ks . This action coincides with the Galois action on the embeddings l → ks of the discriminant extension. By a comparison of the low degree terms of the Leray spectral sequences for X → Spec k and SB(A) → Spec k, we see that the Brauer obstruction to the

Galois invariant class of OXks (1) (which is the sum of the two classes of rulings) is precisely the Brauer class of A. If the discriminant extension is trivial, then both ruling classes are Galois invariant, and each one could have its own Brauer obstruction. Thus for an involution variety X, the low terms of the Leray spectral sequence (see (2.1)) show that ωX generates Pic(X) if and only if the discriminant extension is nontrivial. ∼ 1 1 An involution surface S is an involution variety of dimension 2, i.e., Sks = Pks × Pks . In particular, S is a minimal geometrically rational del Pezzo surface. As intersection numbers do not change under scalar extension, S has degree 8. The set of k-isomorphism classes of involution surfaces is also in bijection with the set of isomorphism classes of pairs (l, B), where l is an étalequadratic extension of k and B is a quaternion algebra over l, see [67, §15.B]. Given an involution variety S corresponding to a central simple algebra (A, σ) of degree 4 with quadratic pair, the even Clifford algebra C0(A, σ) is a quaternion algebra over the discriminant extension. Conversely, given a pair (l, B), the associated involution variety is the Weil restriction S = Rl/kSB(B). As far as placing involution surfaces into the classification of minimal geometrically rational surfaces, there are several cases: If the discriminant extension is trivial, then S belongs ∼ to case (iv). In this case S = C1 ×C2 is a product of Severi–Brauer curves. Writing Ci = SB(Bi) ∼ for quaternion algebras Bi over k, then A = B1 ⊗ B2. In this case, the Brauer class of A is ∼ trivial if and only if C1 = C2. If the discriminant extension is nontrivial, then S belongs to case (ii) or (iii) depending on whether the Brauer class of A is trivial or not, respectively. Example 3.4. Let S be a geometrically rational del Pezzo surface of degree 8 not isomorphic to an involution variety. The unique exceptional curve on Sks is a Galois invariant subvariety, hence can be contracted to arrive at a del Pezzo surface of degree 9 with a rational point, which 2 2 2 is thus Pk. Thus S → Pk is the blow up of Pk at a single k-rational point. In particular, S is a rational del Pezzo surface and is never minimal. 2 Let X be a geometrically rational del Pezzo surface of degree 7. As Sks is the blow-up of Pks at two points, the exceptional divisor consists of a Galois invariant pair of (−1)-curves, hence can be contracted to arrive at a del Pezzo surface of degree 9 with a point of degree 2, which is 2 2 2 thus Pk. Thus S → Pk is the blow up of Pk at a closed point of degree 2. In particular, S is a rational del Pezzo surface and is never minimal. G Remark 3.5. If S is a minimal del Pezzo surface of degree ≤ 6, then Pic(S) → Pic(Sks ) k is an isomorphism, cf. [41, Lemma 2.5, Prop. 5.3]. We continue our general discussion of geometrically rational surfaces. Denote by ρ(S) the Picard rank of S.

Proposition 3.6. If S is a geometrically rational surface over a field k, then K0(S)Q is a Q- vector space of dimension 2 + ρ(S). In particular, if S is minimal then K0(S)Q has dimension 3 or 4, and in the latter case S has a conic bundle structure. Proof. The Chern character M i ch : K0(S) ⊗Z Q → CH (S) ⊗Z Q i 0 ∼ 1 ∼ 2 is an isomorphism. We have CH (S) = Z and CH (S) = Pic(S). Finally, to describe CH (S) = CH0(S), we have an exact sequence deg 0 → A0(S) → CH0(S) −−→ Z → Z/i(S)Z → 0 18 ASHER AUEL AND MARCELLO BERNARDARA where A0(S) is defined to be the kernel of the degree map and i(S) is the index, i.e., the greatest common divisor of degrees of closed points on S. Since S is rational after a finite separable extension (by Proposition 3.1), a restriction-corestriction argument shows that A0(S) is torsion, ' cf. [40, Prop. 6.4]. This implies that the degree map becomes an isomorphism CH0(S)⊗Z Q → Q after tensoring with Q. This completes the calculation of the dimension. The final statement is a result of the classification of minimal geometrically rational surfaces. Corollary 3.7. Let S be a geometrically rational surface over k. If there exist field extensions l1, . . . , ln of k and Azumaya algebras Ai over li, such that there is a semiorthogonal decomposition b b b D (S) = hD (l1/k, A1),..., D (ln/k, An)i, then n = 2 + ρ(S). In particular, S is categorically representable in dimension 0 if and only if there exist field extensions l1, . . . , ln, with n = ρ(S) + 2, and a semiorthogonal decomposition b b b (3.1) D (S) = hD (l1/k),..., D (ln/k)i. Proof. The first statement is a corollary of Proposition 3.6, since the semiorthogonal decomposition gives a splitting n M K0(S) = K0(li,Ai) i=1 ∼ and K0(l, A) = Z for any field l and any Azumaya algebra A over l. To prove the second statement notice that S is categorically representable in dimension zero if and only if a semiorthogonal decomposition like (3.1) exists by Lemma 1.19, and the number of component is given by the first statement. Let us recall a straightforward consequence of Orlov’s result on blow-ups, reducing the question of being categorically representable in dimension 0 to minimal surfaces. Lemma 3.8. Let S be a smooth projective non-minimal surface over k. Then there is a smooth projective minimal surface S0, and a fully faithful functor Φ: Db(S0) → Db(S) such that the orthogonal complement of Φ(Db(S0)) is representable in dimension 0. Proof. Since S is not minimal, there exists a k-birational morphism π : S → S0 to a minimal surface. Then π is the blow-up of a closed zero-dimensional subvariety Z ⊂ S0. The proof follows from Orlov blow-up formula [84]. Manin has proved that, given a (non necessarily minimal) del Pezzo surface of degree d ≥ 2, the existence of a k-rational point (not lying on any exceptional curve if d ≤ 4) implies the 2 existence of a unirational parametrization, i.e., a map Pk → S of finite degree. Theorem 3.9 (Manin [82, Thm. 29.4]). Let S be a del Pezzo surface of degree d ≥ 2 over k with S(k) 6= ∅. If d ≤ 4 suppose moreover that the point does not lie on any exceptional curve. 2 Then there exists a rational map φ : Pk → S whose degree δd is given by the following table d ≥ 5 4 3 2 δd 1 2 6 24 In particular, if d ≥ 5, the surface S has a k-rational point if and only if it is k-rational. Minimal del Pezzo surfaces of degree ≤ 7 can be characterized by the Galois action on exceptional curves. We say that a del Pezzo surface S over k is totally split if S is k-rational and all exceptional curves are defined over k. Any field extension of k over which a del Pezzo surface becomes totally split will be called a total splitting field for S. We can always choose a finite Galois total splitting field for a del Pezzo surface. We remark that S can be split, but not necessarily totally split, over a given field. We end this section by recalling the following classification of birational maps between non- rational minimal del Pezzo surfaces, which can be proved by classifying all the possible links in the Sarkisov program (see [58]). DEL PEZZO SURFACES 19

Proposition 3.10 (Iskovskikh [58, Thms. 1.6, 4.5, 4.6]). Let S be a non rational del Pezzo 0 0 surface of Picard rank 1, S a minimal surface, and φ : S 99K S a k-birational map. (i) If deg(S) = 1 or if S has no closed point x of degree < deg(S), then φ is an isomorphism (i.e., S is birationally rigid). (ii) If deg(S) = 2 and S(k) 6= ∅, or if deg(S) = 3 and S(k) 6= ∅ or if deg(S) = 4 and S has a point of degree 2 and no point of degree 1 or 3, or if S has degree 8 and a point of degree 4 (but no point of lower degree), then φ can be composed with a birational map S 99K S to give an isomorphism (i.e. S is birationally semirigid). (iii) If deg(S) = 6 or deg(S) = 9, then deg(S) = deg(S0) (i.e. S is deg-rigid). (iv) If deg(S) = 4 and S has a k-rational point or if deg(S) = 8 and S has a degree 2 point (but no k-rational point), then the blow-up of S along such a point is a conic bundle of degree 3 or 6, respectively. In particular, S is not deg-rigid. All non rational del Pezzo surfaces of Picard rank one are covered by one of these cases. Proof. A proof of all of these results can be found in [58] (some of them were previously known). Item (i) is [58, Thm. 1.6], while items (ii), (iii), and (iv) are summarized in [58, Comment 5]. Notice that if S has degree 8 and a point of degree 4 but no point of lower degree, then [58, Thm. 4.4] only says that S is deg-rigid. In this case, S is an involution variety, for which the birational semirigidity can be proved directly via the theory of quadratic forms, see Proposi- tion C.3. Finally, the list is exhaustive since if deg(S) ≤ 6 and S has a closed point of degree < deg(S) and coprime with deg(S), then S is rational (for an argument, see [58, p. 624]). Similarly if S has degree 5 or 7 then it is rational.

Part 2. Del Pezzo surfaces The plan of this part is as follows. In Section4 we recall the necessary facts concerning 3-block decompositions. In Section5 we consider the case of degree d ≤ 4 and prove Theorem 1 and3. The rest of the proofs will be divided into the cases of Severi–Brauer surfaces (Section 6), involution surfaces (Section7), del Pezzo surfaces of degree 7 (Section8), 6 (Section9), and 5 (Section 10).

4. Exceptional objects and blocks on del Pezzo surfaces In this section, we assume that k is separably closed. The derived category of a totally split del Pezzo surface (over an algebraically closed field) has been extensively studied by the Moscow school in the 1990’s (see, e.g., [51], [52], [92], [69, 64]), with a particular attention to the structure of exceptional collections. While these authors restrict to working over algebraically closed fields of characteristic zero, their proofs are based on properties of vector bundles and on the description of a del Pezzo 2 surface as a blow-up of P , hence they actually hold for any totally split del Pezzo surface, in particular, over any separably closed field. Let S be a del Pezzo surface over k. Then S is totally split and S is either a quadric surface 2 and has degree 8, or S is a blow-up of P in 9 − d points, for 1 ≤ d ≤ 9, and has degree d.A 3-block decomposition of Db(S) is a semiorthogonal decomposition Db(S) = hE, F, Gi consisting of exceptional blocks. Gorodentsev–Rudakov, Rudakov, and Karpov–Nogin (see [52], [92], [64]) have proved the existence of 3-block decompositions by exceptional vector bundles with some unicity property (up to mutations). Indeed, the ranks and degrees of vector bundles are constant in a block, and there is an algorithm to compute how slopes change under mutations, as explained in [64]. A 3-block decomposition is minimal if any mutation increases the rank of one of the blocks. The existence of minimal decomposition is related to solutions of Markov-type equations. 20 ASHER AUEL AND MARCELLO BERNARDARA

E F G deg(S) n r exc. collection n r exc. collection n r exc. collection 9 1 1 O 1 1 O(H) 1 1 O(2H) O(1, 0) 8 (inv) 1 1 O 2 1 1 1 O(1, 1) O(0, 1) 8 (dP) No 3-block decomposition 7 O(H) O(2H − L1 − L2) 6 1 1 O 2 1 3 1 O(2H − L1 − L3) O(2H − L≤3) O(2H − L2 − L3) O(H) O(L1 − ω − H) 5 1 1 O 1 2 F 5 1 O(L2 − ω − H) O(L3 − ω − H) O(L4 − ω − H) (H) −ω (L ) O O 4 (2H − L − L ) 4 2 1 2 1 4 1 O 1 2 O(2H − L1 − L3) O(L5) O(2H − L≤3) O(2H − L2 − L3) O(L4) O(H − L1) −ω 3 (i) 3 1 O(L5) 3 1 O(H − L2) 3 1 O(H) O(L6) O(H − L3) O(2H − L≤3) O(H) O(L1 − ω) O(L2 − ω) O(L3 − ω) 3 (ii) 1 2 T6 2 1 6 1 −ω O(L4 − ω) O(L5 − ω) O(L6 − ω) −ω 2 (i) 1 2 E7 1 2 T7 8 1 O(H − L1) ... O(H − L7) O(L4) −ω E7 O(L5) O(H − L1) 2 (ii) 2 2 0 4 1 4 1 E7 O(L6) O(H − L2) O(L7) O(H − L3) O(H − L1) O(H) 00 2 (iii) 1 3 E7 3 1 O(H − L2) 6 1 O(2H − L≤3) O(H − L3) O(L4 − ω) ... O(L7 − ω) −ω 1 (i) 1 3 E8 1 3 F8 9 1 O(−L1) ... O(−L8) 0 T8 O(L4 − ω) ... O(L8 − ω) 1 (ii) 1 4 E8 2 2 0 8 1 T8 O(H − L1) O(H − L2) O(H − L3) 00 T8 E7 O(L4 − ω) O(L5 − ω) O(L6 − ω) 0 1 (iii) 2 4 3 2 T8 6 1 00 E8 00 O(H − L1) O(H − L2) O(H − L3) T8 O(H) 000 1 (iv) 1 5 E8 5 2 F4,8 ...F8,8 5 1 O(2H − L≤3) O(H − L1 − ω) ... O(H − L3 − ω) Table 1. Representative sets of exceptional objects generating the various 3- b block decompositions of D (Sks ), taken up to mutation, tensoring through by a line bundle, and the Weyl group action. For each block n = number of bundles 2 in the block and r = rank of these bundles. Here: S → Pks is the blow-up at 9 − deg(S) points, Li the exceptional divisors, and H the pull back of the P P hyperplane class, L≥4 := i≥4 Li, and L≤3 := i≤3 Li. For the higher rank vector bundles, we follow the notation in Karpov–Nogin [64, §4]. DEL PEZZO SURFACES 21

Proposition 4.1 ([52], [92], [64]). Let S be either a quadric surface or a del Pezzo surface of degree d 6= 7, 8 over a separably closed field k. Set s = max{1, 5 − d}. Then Db(S) has s (up to tensoring by line bundles and the action of the Weyl group) minimal 3-block decompositions such that any other 3-block decomposition of Db(S) can obtained from one of these by a finite number of mutations. In all cases, the blocks are generated by a completely orthogonal set of vector bundles, so that each block has a tilting bundle. In the cases of degree d = 7, 8 (see Example 3.4), there is always a 4-block decomposition, but not a 3-block one. On the other hand, del Pezzo surfaces of these degrees are never minimal (see Example 3.4). We summarize the possible minimal 3-block decompositions in Table1. Recall that these decompositions hold over a separably closed field, or in general for totally split del Pezzo surfaces, 1 1 2 so that S is either Pk × Pk or the blow up of P in 9 − d rational points, and exceptional means k-exceptional. In the first case we use the standard notation O(a, b) for line bundles of bidegree 2 (a, b). In the latter, we denote by Li (for i = 1, . . . r) the exceptional divisors of S → P , and 2 by H the pull-back of the hyperplane class in P . For each block, the table lists the number n of exceptional bundles, their rank r (which is constant within the block), and the 1st Chern class of the tilting bundle of the block (i.e., the sum of the 1st Chern classes of the bundles in the block).

5. Del Pezzo surfaces of Picard rank 1 and low degree From now on, let k be an arbitrary field and S be a del Pezzo surface of degree d and Picard ⊕3 rank 1 over k. Recall that K0(S)Q ' Q , so that we can wonder, according to Proposi- tion 3.6, whether there is a semiorthogonal decomposition given by three simple k-algebras. b If such a decomposition exists, it must base change to a 3-block decomposition of D (Sks ) by Proposition 2.12. Conversely, if we suppose that there is a semiorthogonal decomposition Db(S) = hE, F, Gi whose base change is a 3-block decomposition, then Proposition 2.12 guarantees that the three components are equivalent to derived categories of simple k-algebras. Combining the explicit description of the vector bundles forming exceptional blocks on Sks , together with the previous observations, we gain control over semiorthogonal decompositions of derived categories of del Pezzo surfaces of Picard rank 1. Theorem 5.1. Let S be a del Pezzo surface of degree d ≤ 4. Then there is no semiorthogonal decomposition b b b b (5.1) D (S) = hD (l1/k, α1), D (l2/k, α2), D (l3/k, α3)i, with li field extensions of k and αi in Br(li). Proof. If a decomposition (5.1) exists, then ρ(S) = 1 by Corollary 3.7, hence Pic(S) = Z[ω] by the classification. Moreover, Proposition 2.12 ensures us that its base change to ks is a 3-block decomposition. Up to mutating the three blocks, tensoring with line bundles, and the action of the Weyl group, we can appeal to the Karpov–Nogin classification (cf. Proposition 4.1) and proceed in a case-by-case analysis. Such exceptional blocks are generated by vector bundles, so Corollary 2.13 guarantees that a nontrivial Z-linear combination of the first Chern classes of these vector bundles is a multiple of ω. ⊥ First, the Galois action on hωi ⊂ Pic(Sks ) factors through the Weyl group of the associated root system, see [44, Thm. 2]. Hence over ks , there is a choice of 9 − d pairwise disjoint exceptional lines L1,...,Ld so that the three blocks are described as in Table1, up to tensoring all exceptional bundles by the same line bundle O(M) in Pic(Sks ). Our argument will follow two different paths, depending on the degree and subcase: (1) Either we show that one of the blocks contains a proper admissible subcategory generated by ω, and then Lemma 1.16 shows that this contradicts ρ(S) = 1. 22 ASHER AUEL AND MARCELLO BERNARDARA

(2) Or we show the impossibility of descending a non-trivial generator of one of the blocks, by proving that its first Chern class could never be a multiple of ω. Pd In what follows, we will consider a line bundle M on Sks written as M = nH + i=1 aiLi. Pd Tensoring by powers of ω = −3H + i=1 Li, we can choose to fix one of the coefficients ai or a representative of n modulo 3. s Degree 4. In degree 4, E is generated by O(L4) and O(L5) over k . Let O(M) be a line bundle on Sks . If there exists a pair of integers a and b such that

(5.2) a(L4 + M) + b(L5 + M) = rω, then r = 0. Indeed, we make this ﬁrst calculation explicit: let us assume that M = nH + P5 i=1 aiLi with a1 = 0. Then it follows that r = 0, in which case, we must also have n = 0 and a2 = a3 = 0. Of course, we can assume that both a and b are not both zero, otherwise no nontrivial generator of the block E descends. In fact, since Pic(Sks ) is torsion-free, we can further take a and b to be coprime. With this in mind, equation (5.2) yields

a(L4 + a4L4 + a5L5) + b(L5 + a4L4 + a5L5) = 0.

It follows, looking at the coeﬃcients of L4 and L5, respectively, that a + (a + b)a = 0 (5.3) 4 b + (a + b)a5 = 0 Since a and b are coprime, a + b 6= 0 and hence the only integer solutions to this system of equations has a4 = a5 = 0 (which implies a = b = 0, a contradiction) or a + b = ±1. Suppose that a + b = 1. From (5.3), we get that a4 = −a and a5 = a − 1, so that the possibilities to descend the block E are obtained by tensoring the Karpov–Nogin 3-block decomposition over Sks by M = −aL4 + (a − 1)L5, for some integer a. Now we consider the block F. After tensoring with O(M), the block F is generated, over ks , by O(H + M) and O(2H − L1 − L2 − L3 + M). If the block descends to k, we have integers α, β, and ρ such that

(5.4) α(H + M) + β(2H − L1 − L2 − L3 + M) = ρω.

Looking at coefficients of L1 in (5.4), we get β = −ρ. Then considering the coefficient of H in (5.4) gives α = −ρ. Finally, the coefficients of L4 and L5 give αρ = ρ and (1 − α)ρ = ρ, respectively. From this, it follows that ρ = 0, so that α = β = 0, hence no nontrivial generator of the block F can descend. If we suppose that a + b = −1, then M = aL4 − (a + 1)L5, and similar arguments show that no nontrivial generator of F can descend. Using Corollary 2.13, this means that there is no way to descend both the blocks E and F to k at the same time.

Degree 3, case (i). The block E is generated by O(L1), O(L2), and O(L3). We consider the equation a(L1 + M) + b(L2 + M) + c(L3 + M) = rω. P6 As before, up to tensoring by multiples of ω, we can assume that M = nH + i=1 aiLi with a4 = 0, from which we similarly conclude that r = 0, hence c1(E) = 0, and n = a4 = a5 = a6 = 0. The block F is generated by O(H − L4), O(H − L5), O(H − L6), so that we are looking for nontrivial integers α, β, and γ such that

α(H − L4 + M) + β(H − L5 + M) + γ(H − L6 + M) = ρω where M = a1L1 + a2L2 + a3L3, as just shown. From this, we get α + β + γ = −3ρ (considering the coefficients of H) and α + β + γ = −ρ (considering the coefficients of L4). It follows that ρ = 0, and hence c1(F) = 0. Notice that the latter does not mean that the coefficients α, β, and γ are trivial, so we can’t appeal to Corollary 2.13 here. However, looking at the coefficients of L1, L2 and L3, we also get a1 = a2 = a3 = 0, from which we have that M is a multiple of ω. Over ks , the block G contains ω as part of an exceptional sequence (according to Table1), hence after tensoring by M, the block G contains a multiple ω⊗m for some integer m. The category generated by this ω⊗m is a proper admissible subcategory of G. If G descends to an DEL PEZZO SURFACES 23 admissible subcategory Db(K,A) of Db(S), then this subcategory admits hω⊗mi as a proper admissible subcategory. Then Lemma 1.16 shows that K cannot be a field.

Degree 3, case (ii). The block E is generated by a rank 2 vector bundle T6 with c1(T6) = H. We have c1(T6 ⊗ M) = H + 2M, and for the block to descend, we need H + 2M = rω. Taking M (up to multiples of ω) with a1 = 0, we conclude that r = 0. As a result, ai = 0 for all i, so that M = nH is a multiple of H, and then H + 2M = (2n + 1)H. We conclude that the only multiple of T6 that can descend is the trivial one. This contradicts Corollary 2.13. Degree 2, case (i). This case is very similar to the case of Degree 3, case (ii). Indeed, the block F is generated by a single rank 2 vector bundle with ﬁrst Chern class H.

Degree 2, case (ii). In this case, F is generated by O(L4),..., O(L7). Arguing as in the case of Degree 4, we deduce that if there exists a linear combination of L4 + M,...,L7 + M (a P7 necessary condition for F to descend) then r = 0 and M = i=4 aiLi. The block E is generated by two rank 2 vector bundles of ﬁrst Chern classes H + ω and −H + L≥4, respectively. As a necessary condition for E to descend, we are looking for a pair of integers a and b such that

a(H + ω + 2M) + b(−H + L≥4 + 2M) = rω.

Since M = a4L4 + ... + a7L7, considering the coefficient of L1, we arrive at a = r, from which it follows that r(H + 2M) + b(−H + L≥4 + 2M) = 0, by subtracting rω on both sides. Now, looking at the coefficient of H, we arrive at b = r, hence we conclude that r(4M + L≥4) = 0. Once again considering the coefficient of H, we see that r 6= 0 is impossible. It follows that a = 0 and hence b = 0, as well. This contradicts Corollary 2.13. Degree 2, case (iii). In this case, the block E is generated, over ks , by a rank 3 vector bundle with first Chern class L≥4. A necessary condition for E to descend is that there exists an integer a such that a(L≥4 + 3M) = rω. Taking M with a1 = 0, we easily get r = 0 and M = a4L4 + ... + a7L7. As in the case of Degree 2, case (ii) we find a contradiction to Corollary 2.13. Degree 1, case (i). This case is similar to that of Degree 3, case (ii). Indeed, in both cases the block E is generated by a single vector bundle. In this case, its first Chern class is −H + 2ω. Arguing as in the previous case, we arrive at a contradiction to Corollary 2.13. Degree 1, case (ii). In this case the block E is generated by a single vector bundle of rank 4 and first Chern class 2H − L1 − L2 − L3. Modifying M by multiples of ω, we can choose a4 = 0, and we can then conclude that a(2H − L1 − L2 − L3 + 4M) = rω implies that r = 0 P7 and M = nH + i=4 aiLi. Looking at the coefficients of H, we see that if a 6= 0, there is no n such that the latter equation holds. Hence a = 0 and we find a contradiction to Corollary 2.13. Degree 1, case (iii). In this case, the block E is generated by two rank 3 vector bundles with first Chern classes L4 + L5 + L6 + L7 and L4 + L5 + L6 + L8, respectively. Hence a necessary condition for E to descend is that there exist integers a and b such that

a(L4 + L5 + L6 + L7 + 3M) + b(L4 + L5 + L6 + L8 + 3M) = rω. As before, we can assume that a and b are both nonzero and coprime. Choosing M = nH + P8 i=1 aiLi with a1 = 0, we arrive at r = 0 (by considering the coeﬃcient of L1), and hence also n = a1 = a2 = a3 = 0. Now, looking to the coeﬃcients of L7 and L8, we arrive at a system of equations a + 3(a + b)a7 = 0 b + 3(a + b)a8 = 0 However, this system has no integer solutions when a and b are coprime, as seen by reducing modulo 3. This contradicts Corollary 2.13. Degree 1, case (iv). In this case, E is generated by a single rank 5 vector bundle with Chern class −2ω + L≥4. Similarly as in the case Degree 3, case (ii), the descent of E yields a contradiction to Corollary 2.13. 24 ASHER AUEL AND MARCELLO BERNARDARA

Thus in each case of degree ≤ 4, the assumptions that all three blocks can descend to simple categories leads to a contradiction and the proof is complete. Corollary 5.2. Let S be a del Pezzo surface of degree d ≤ 4 and Picard rank one. Then S is not categorically representable in dimension 0. Proof. By Corollary 3.7, categorical representability of S would be given by a semiorthogonal decomposition as (5.1) with αi = 0.

On the other hand, given any geometrically rational surface S, the line bundle OS (or the b line bundle ωS) deﬁnes an exceptional object in D (S), hence we always have a nontrivial semiorthogonal decomposition. Proposition 5.3. Let S be a geometrically rational surface over k. Then there is a semiorthogonal decomposition b b D (S) = hD (k), ASi. ⊕2 If ρ(S) = 1, then K0(AS)Q = Q . Furthermore, if S is a del Pezzo surface of degree ≤ 4, then there is no semiorthogonal decomposition b b AS = hD (l1/k, α1), D (l2/k, α2)i with li ﬁelds and αi in Br(li). b Proof. The admissible subcategory D (k) is generated by the exceptional object OS, hence the semiorthogonal decomposition exists. The calculation of K0(AS) is straightforward, and the last statement is a consequence of Theorem 5.1. Proposition 5.4. Let S be a del Pezzo surface of degree d ≤ 4 and Picard rank 1, and suppose that, if d = 4, then ind(S) > 1. Let S0 be birational to S. Then there is a semiorthogonal decomposition AS0 = hT, ASi, where T is representable in dimension 0, and T = 0 if and only if S ' S0. Proof. Under the assumptions, S is birationally rigid. Hence, if S0 is minimal, S0 ' S, and if 0 0 S is not minimal, there is a blow-up S → S. Then we conclude by the blow-up formula. Remark 5.5. If S is a del Pezzo surface of degree ≤ 4, notice from Table1 that there is no 3 block decomposition with a block generated by a single exceptional line bundle. It follows that the category AS does not base-change to a category generated by two exceptional blocks.

We end this section with a conjecture on the structure of the category AS for a del Pezzo surface of degree ≤ 4. The highly non-trivial noncommutative structure of AS should be a reﬂection of the more complicated arithmetic behavior of S.

Conjecture 5.6. If S is a del Pezzo surface of degree ≤ 4 and ρ(S) = 1, then the category AS has no nontrivial semiorthogonal decomposition.

We now describe the few known facts about AS in degrees 3 and 4. If S has degree 4, then 4 the anticanonical embedding S → Pk realizes S as the intersection of two quadric hypersurfaces. The following result was shown in [11], as a consequence of Homological Projective Duality (see also [71]). 1 Theorem 5.7. Let S be a del Pezzo surface of degree 4 and X → P be the associated pencil 4 1 of quadrics of P containing S, with associated even Clifford algebra C0 over P . Then there is a semiorthogonal decomposition b b b 1 D (S) = hD (k), D (P , C0)i. 1 If S(k) 6= ∅ then the quadric threefold fibration X → P can be reduced by hyperbolic 1 0 splitting (cf. [11, §1.3]) to a conic bundle Y → P with Clifford algebra C0, and there is an b 1 b 1 0 equivalence between D (P , C0) and D (P , C0) by [11, Thm. 3]. We will see another (more explicit) way to describe the orthogonal complement AS via a conic bundle in AppendixB. 3 If S has degree 3, then the anticanonical embedding S → Pk realizes S as a cubic hypersurface, so that we can use Kuznetsov’s calculation (see [72]). DEL PEZZO SURFACES 25

S ind(S) A1 Z ind c2 (V1)ks A2 Z ind c2 (V2)ks SB(A) 3 A k 3 3 O(1)⊕3 A−1 k 3 12 O(2)⊕3 2 Pk 1 k k 1 0 O(1) k k 1 0 O(2) Table 2. The invariants of a Severi–Brauer surface S (a del Pezzo surface of degree 9). Here, the algebras Ai = End(Vi) are listed up to Morita equivalence; Z and ind refer to the center and index of Ai; and c2 refers to the 2nd Chern class of Vi.

Theorem 5.8. If S is a del Pezzo surface of degree 3, there is a semiorthogonal decomposition b b D (S) = hD (k), ASi, 4 with AS a Calabi–Yau category of dimension 3 .

The category AS can be described via Matrix Factorization as in [86, Thm. 40] and via Homological Projective Duality as in [15].

6. Severi–Brauer surfaces (del Pezzo surfaces of degree 9) If S is a del Pezzo surface of degree 9 over k, then S is the Severi–Brauer surface associated to a degree 3 central simple algebra A over k. Denote by α ∈ Br(k) the Brauer class of A. Proposition 6.1. Let S = SB(A) be a Severi–Brauer surface. Then the following are equivalent: (i) S has a k-point, (ii) S is k-rational, (iii) S is categorically representable in dimension 0, (iv) AS is categorically representable in dimension 0. Proof. It’s a result of Châtelet(cf. Example 3.2) that (i) is equivalent to (ii) is equivalent to ∼ 2 S = P is equivalent to the triviality of A in the Brauer group. In turn, this implies (iii) and (iv) b 2 by considering the full exceptional collection {O, O(1), O(2)} of D (P ) described by Beilinson [17] and Bernˇste˘ın–Gelfand–Gelfand[24]. It then suffices to prove that (iii) implies that A has trivial Brauer class. In general, a semiorthogonal decomposition (6.1) Db(S) = hDb(k), Db(k, A), Db(k, A−1)i b 2 was constructed in [19], which base changes to the semiorthogonal decomposition D (Pks ) = hO, O(1), O(2)i, see Example 2.14. Assuming (iii), then by Corollary 3.7, there are fields li and a semiorthogonal decomposition b b b b (6.2) D (S) = hD (l1/k), D (l2/k), D (l3/k)i, which by Proposition 4.1, base changes to a 3-block exceptional collection, unique up to mutation and tensoring by line bundles on S. Hence, up to mutations, the decomposition (6.2) base b 2 changes to the decomposition D (Pks ) = hO(i), O(i + 1), O(i + 2)i. Twisting by powers of the canonical bundle and performing one more mutation, we can assume that i = 0. Hence the decomposition (6.2) is equivalent to the decomposition (6.1). In particular, we get that b b D (k, A) ' D (li/k) for some i = 1, 2, which by Corollary 1.18, implies that li = k and that A is split.

Now we verify that the Griffiths–Kuznetsov component GKS is well defined. Proposition 6.2. Let S = SB(A) be a nonrational Severi–Brauer surface. Then the Griffiths– Kuznetsov component is a well defined birational invariant in the following sense: if S1 99K S is a birational map then there is a semiorthogonal decomposition b b b −1 D (S1) = hT, D (k, α), D (k, α )i where T is representable in dimension 0. 26 ASHER AUEL AND MARCELLO BERNARDARA

Proof. If S1 is minimal then S1 = SB(B) is a Severi–Brauer surface by Proposition 3.10. Amit- sur’s theorem [2] implies that B = A or B = A−1. Indeed, in the Appendix (see Proposi- tion A.1), we can show how the decomposition of the birational map S1 99K S gives either Db(k, A) ' Db(k, B) or Db(k, A) ' Db(k, B−1) only using the description of tilting bundles and their behavior under birational maps. Then the statement follows, possibly up to a mutation, from the semiorthogonal decomposition (6.1). If S1 is not minimal, there is a minimal model S1 → S0, and we conclude using Lemma 1.15 and the ﬁrst part of the proof. Remark 6.3. Recall the vector bundle V of rank 3 on S constructed by Quillen [91, §8.4] (see ⊕3 Example 2.14) such that Vks = O(1) . More explicitly, if K/k is a separable degree 3 extension min ∨ ∨ min splitting A, then Vi = trK/kO 2 (i) by Theorem 2.9. We set V1 = V and V2 = (V ⊗ ω ) . PK S ⊕3 Remark that V2,ks is either O(2) or O(2) . These vector bundles are tilting bundles for the blocks F and G, respectively, and A1 = End(V1) is Morita equivalent to A, A2 = End(V2) is −1 Morita equivalent to A , and and Vi are indecomposable. We list the ranks and 2nd Chern classes of the vector bundles Vi on Table2. The calculation of the second Chern classes of the vector bundles V1 and V2 is easily obtained s by their description over k . In particular, we notice that ind(S) = gcd(c2(V1), c2(V2)) when S ⊕2 ⊕2 is nonsplit and gcd(c2(V1 ), c2(V2 )) = ind(S). Remark 6.4. The second Chern classes of the generators of Db(k, A) and Db(k, A−1) are not stable under mutations. The values 3 and 12 are obtained for the speciﬁc choices of bundles listed in Table2. This same remark applies to all other degrees.

7. Involution varieties (del Pezzo surfaces of degree 8) If the degree of S is 8, either S is an involution surface (cf. Example 3.3), or S is the blow-up 2 of Pks at a k-rational point (cf. Example 3.4). In the latter case, S is rational, not minimal, and has a semiorthogonal decomposition b b b b b D (S) = hO, O(H), O(2H), O(L1)i = hD (k), D (k), D (k), D (k)i 2 where H is the pull back of the hyperplane class from Pk to S and L1 is the exceptional divisor of the blow-up. There is no 3-block decomposition in this case. So we focus our attention on involution surfaces. In this case, S is associated to a degree 4 central simple k-algebra (A, σ) with quadratic pair. The even Clifford algebra C0(A, σ) (see Example 3.3) is a quaternion algebra over its center l, which is the étalequadratic discriminant extension of S. Denote by α the Brauer class of A and γ the Brauer class of C0(A, σ) over the 2 2 discriminant extension l. When l = k , we write γ = (γ+, γ−) ∈ Br(k ) = Br(k) × Br(k). The fundamental relations for the Clifford algebra of an algebra with involution [67, Thm. 9.14] imply that α = corl/kγ ∈ Br(k). We record another important fact from the algebraic theory of quadratic forms.

Lemma 7.1. Let S be an involution surface over a ﬁeld k. Then S(k) 6= ∅ if and only if γ ∈ Br(l) is trivial.

Proof. If S(k) 6= ∅ then SB(A)(k) 6= ∅, hence α is split. Also, if γ ∈ Br(l) is trivial, then α ∈ Br(k) is trivial by the fundamental relations. Thus we can reduce to the case when (A, σ) is adjoint to a quadratic form q of dimension 4 over k, which in this case q is isotropic if and only if C0(q) is split over its center, i.e., γ ∈ Br(l) is trivial, by [68, Thm. 6.3] (also see [94, 2 Thm. 14.1, Lemma 14.2] in characteristic 6= 2 and [14, II Prop. 5.3] in characteristic 2). Proposition 7.2. Let S be an involution surface over a ﬁeld k. Then the following are equivalent: (i) S has a k-point, (ii) S is k-rational, (iii) S is categorically representable in dimension 0, (iv) AS is categorically representable in dimension 0. DEL PEZZO SURFACES 27

S ind(S) ρ(S) A1 Z ind c2 rk A2 Z per ind c2 rk

8.1 S ⊂ SB(A) 4 1 C0 l 2 4 4 A k 2 4 12 4 8.2 SB(B) × SB(B0) 4 2 B × B0 k2 2 4 4 B ⊗ B0 k 2 4 12 4 8.3 S ⊂ SB(A) 2 1 C0 l 2 4 4 A k 2 2 2 2 3 8.4 S ⊂ Pk 2 1 C0 l 2 4 4 k k 1 1 0 1 8.5 SB(B) × SB(B0) 2 2 B × B0 k2 2 4 4 B ⊗ B0 k 2 2 2 2 8.6 SB(B) × SB(B) 2 2 B × B k2 2 4 4 k k 1 1 0 1 8.7 SB(B) × P1 2 2 B × k k2 2 4 4 B k 2 2 2 2 3 8.8 S ⊂ Pk 1 1 l l 1 1 2 k k 1 1 0 1 8.9 P1 × P1 1 2 k2 k2 1 1 2 k k 1 1 0 1 Table 3. The invariants of an involution surface S (a minimal del Pezzo of degree 8). Here: the algebras End(V1) and End(V2) are Morita equivalent to A1 = C0 and A2 = A; also Z, per, and ind refer to the center, period, and index of Ai; and c2 and rk refer to the 2nd Chern class and rank of Vi. Note that (V1)ks is a direct sum of spinor bundles while (V2)ks is a direct sum of hyperplane classes. Also l stands for the separable quadratic discriminant extension of k. Recall that S is rational if and only if ind(S) = 1. In all these cases, S is minimal. We give a brief geometric description of all cases in Remark 7.4.

Proof. Our aim is to produce a semiorthogonal decomposition that base changes to the 3-block decomposition from Table1. In characteristic 6= 2, this is a result of Blunk [26, §7], who works with tilting bundles and does not explicitly mention semiorthogonal decompositions. We will give a sketch of an alternate proof that works in any characteristic. First, under the closed embedding S ⊂ SB(A) = X, we get V2 as the pull back of the indecomposable vector bundle on X of pure type OP3 (1). Then V2 has rank dividing 4 and End(V2) is Morita equivalent b to A. Second, the fully faithful embedding of D (k, C0) can be seen as the twisted version of Kuznetsov’s result [71] (see [11, Thm. 2.2.1] for the case of a quadric over a general ﬁeld). More explicitly, OSks (1, 0) ⊕ OSks (0, 1) is a Galois invariant vector bundle, with the Galois group of l/k acting by switching the factors (when the discriminant is nontrivial), and there is a unique s indecomposable vector bundle V1 of this pure type by §2.2. Hence over k , by comparing with the usual decomposition (cf. [71, Lemma 4.14], which is non other than the 3-block decomposition in Table1), we ﬁnd a semiorthogonal decomposition b b b b (7.1) D (S) = hD (k), D (k, C0), D (k, A)i.

The fact that the endomorphism algebra of OSks (1, 0) ⊕ OSks (0, 1) is Morita-equivalent to the even Cliﬀord algebra C0 goes back to Kapranov [60, §4.14]. Now we proceed with the proof of the equivalences. It’s a classical result (cf. Example 3.3) that (i) is equivalent to (ii). By Lemma 7.1, condition (i) is equivalent to the triviality of γ ∈ Br(l) (and also α ∈ Br(k)). In particular, (i) implies (iii) and (iv), since then the semiorthogonal decomposition just constructed is of the form Db(S) = hDb(k), Db(l), Db(k)i, with the ﬁrst block generated by OS. Hence both S and AS is categorically representable in dimension 0 by Lemma 1.19. Finally, we are left to proving that (iii) or (iv) implies the triviality of γ ∈ Br(l). First, assume that S has Picard rank 1. If S is categorically representable in dimension 0, then by Corollary 3.7, there is a semiorthogonal decomposition b b b b (7.2) D (S) = hD (l1/k), D (l2/k), D (l3/k)i, which by Proposition 4.1, base-changes to a 3-block exceptional collection. Hence, up to muta- b tion, tensoring by line bundles on S, and the Weyl group action, we can assume that D (l1/k) b base changes to hOi and D (l3/k) base changes to hO(1, 1)i. Hence the decomposition (7.2) b b base changes to the decomposition (7.1). In particular, we get that D (k, C0) ' D (l2/k) and b b D (k, A) ' D (l3/k), which by Corollary 1.18, implies that l = l2 and C0 is split over l and that A is split over l3 = k. 28 ASHER AUEL AND MARCELLO BERNARDARA

Second, assume that S has Picard rank 2. In this case, we have S = C ×C0 for Severi–Brauer 0 0 0 0 curves C = SB(B) and C = SB(B ), and then C0 = B × B and A = B ⊗ B , cf. Example 3.3. Hence we have a semiorthogonal decomposition b b 0 b (7.3) AS = hD (k, B), D (k, B ), D (k, A)i.

If AS is representable in dimension 0, then by Corollary 3.7, there is a semiorthogonal decomposition b b b (7.4) AS = hD (l1/k), D (l2/k), D (l3/k)i,

Since the Picard rank of S is stable under base change, it follows that li = k for i = 1, 2, 3. Thus b AS is generated by three k-exceptional objects, hence D (S) is generated by four k-exceptional objects. Over ks , we can appeal to Rudakov [92] to mutate the semiorthogonal decomposition (7.4) into the decomposition (7.3). Using Theorem 1.17, we deduce that the Brauer classes of 0 A, B, B , and hence also C0, are trivial. We now want to show that the Griffiths–Kuznetsov component is well defined, except possibly when S has index 2 and Picard rank 1. In the AppendixB, we will see how this case should be thought of as a conic bundle of degree 6 from the categorical point of view. Proposition 7.3. Let S be an involution surface. Then the Griffiths–Kuznetsov component GKS is well defined as a birational invariant in the following cases. Letting S1 99K S be a birational map, we have: • ind(S) = 4 if and only if α ∈ Br(k) has index 4; there is a semiorthogonal decomposition b b AS1 = hT, D (l, γ), D (k, α)i • ind(S) = 2 and ρ(S) = 2 then γ is never trivial; if α is trivial then there is a semiorthog- b onal decomposition AS1 = hT, D (l, γ)i and if α is nontrivial then there is a semiorthog- b b onal decomposition AS1 = hT, D (l, γ), D (k, α)i where T always denotes a category representable in dimension 0.

Proof. By Lemma 7.1, γ is trivial if and only if S(k) 6= ∅. Furthermore, we can always find 0 0 0 a quadratic extension l /l that splits C0 over l. Hence S(l ) 6= ∅. Since l /k has degree 4, it follows that S has a closed point of degree 4 so that ind(S) divides 4. Since S ⊂ SB(A), we have that ind(S) must be a multiple of ind(SB(A)) = ind(A). In particular, ind(A) = 4 if and only if ind(S) = 4. Also if ind(A) = 2, then a generalization of Albert’s result on common splitting fields for quaternion algebras, cf. [67, Cor. 16.28], shows that there is a quadratic extension of k splitting C0, hence also A by the fundamental relations, and thus ind(S) = 2. Now suppose that ind(S) = 4. Then both γ and α are nontrivial. If ρ(S) = 1 then S is birationally rigid by Proposition 3.10. If ρ(S) = 2, then S =∼ SB(B) × SB(B0) such that 0 0 A = B ⊗ B has index 4 and C0 = B × B . In §C, we show that S1 admits a birational 0 morphism S1 → S0 where S0 is a conic bundle over either SB(B) or SB(B ) and has the required semiorthogonal decomposition. Now suppose that ind(S) = 2 and ρ(S) = 2, then S1 admits a birational morphism S1 → S0 where S0 is a conic bundle of degree 8, and in the Appendix §C, we show that it has the required semiorthogonal decomposition. Remark 7.4. We now describe the geometry of the all possible cases listed in Table3. Any involution surface is minimal. If ρ(S) = 1 (equivalently, l is a field) then Pic(S) is generated 3 either by the anticanonical bundle or its square root. In the second case, ind(S)|2 and S ⊂ P is a quadric surface, which can either be isotropic (case 8.8) or anisotropic (case 8.4). If the Picard group is generated by the anticanonical bundle, then S ⊂ SB(A) is a degree 2 divisor of a Severi–Brauer threefold and ind(S) = ind(A) (cases 8.1 and 8.3). In the case when ρ(S) = 2 then S is isomorphic to a product of Severi–Brauer curves SB(B) and SB(B0). In this case, 0 0 0 C0 = B × B and A = B ⊗ B and the possible cases are: B not equivalent to B and both nontrivial (cases 8.2 and 8.5 according to the index of A); B = B0 nontrivial (case 8.6); and B DEL PEZZO SURFACES 29 nontrivial and B0 trivial (case 8.7); and both B and B0 trivial (case 8.9). We remark that cases 8.6 and 8.7 are k-birational to each other for any given B. ⊕2 Remark 7.5. There is a natural vector bundle W of rank 4 on S such that Wks = O(1, 0) ⊕ ⊕2 ⊕4 O(0, 1) . There is also a natural rank 4 vector bundle U on SB(A) such that Uks = O(1) . min min We let V1 = W and V2 = U|S (recall Definition 2.10). These vector bundles are tilting bundles for the blocks F and G, respectively, and have the following properties: A1 = End(V1) is Morita equivalent to C0 and A2 = End(V2) is Morita equivalent to A; V1 is indecomposable if and only if l is a field; and V2 is always indecomposable. We list the ranks and 2nd Chern classes of the vector bundles Vi on Table3. The calculation of the second Chern classes of the vector bundles V1 and V2 is easily obtained s by their description over k . In particular, we notice that ind(S) = gcd(c2(V1), c2(V2)), except ⊕2 when S is an anisotropic quadric surface, in which case gcd(c2(V1), c2(V2 )) = ind(S). Here, we use the convention that gcd(a, 0) = a.

8. del Pezzo surfaces of degree 7 2 A del Pezzo surface S of degree 7 is the blow up of Pk along a point of degree 2, see Ex- ample 3.4. If the residue ﬁeld of the center of blow up is l, then there is a semiorthogonal decomposition Db(S) = hO, O(H), O(2H),V i = hDb(k), Db(k), Db(k), Db(l)i 2 where H is the pull back of the hyperplane class from Pk to S, and V is a rank 2 vector bundle s on S such that End(V ) = l and V ⊗ k = O(L1) ⊕ O(L2), where L1 and L2 are the exceptional divisors on ks . In particular, S is k-rational and is categorically representable in dimension 0 by Lemma 1.15. However, there is no 3-block decomposition of S.

9. del Pezzo surfaces of degree 6 Let S be a del Pezzo surface of degree 6. There is an associated quaternion Azumaya algebra Q over a cubic étale k-algebra L and an associated degree 3 Azumaya algebra B over a quadratic étale k-algebra K. Blunk [25] gives an interpretation of these algebras in terms of a toric presentation of S, building on the geometric construction of Colliot-Thélène,Karpenko, and Merkurjev [41]. Let κ ∈ Br(L) and β ∈ Br(K) denote the Brauer classes of Q and B, respectively. Blunk, Sierra, and Smith [27] provide a semiorthogonal decomposition (9.1) Db(S) = hDb(k), Db(k, Q), Db(k, B)i = hDb(k), Db(L/k, κ), Db(K/k, β)i, Our first task is to prove that the semiorthogonal decomposition (9.1) descends the minimal three block decomposition from [64]. This will give an alternative description of the tilting bundles generating the blocks. Proposition 9.1. The base change of the semiorthogonal decomposition (9.1) coincides with b the minimal 3-block decomposition of D (Sks ). Remark 9.2. We can appeal to Proposition 4.1 for the fact that the semiorthogonal decomposition (9.1) coincides, up to mutation, tensoring by a line bundle, and the Weyl group action, with the minimal 3-block decomposition. Hence in Proposition 9.1, we prove slightly more: using Blunk’s work [25], we explicitly describe the generators of the semiorthogonal components of (9.1) that base change to the 3-block collection from Table1. Aside from being necessary for the sequel, we believe that the direct proof clarifies the connection between the beautiful geometry and arithmetic of del Pezzo surfaces of degree 6 and its derived category. Before giving the proof, we recall the construction in Blunk [25], and Blunk, Sierra, and s Smith [27], of certain vector bundles on S. The del Pezzo surface Sks of degree 6 over k 2 is the blow-up of Pks in three noncolinear points p1, p2, p3. There are six exceptional lines, coming in two pairs of three lines, say L1,L2,L3 and M1,M2,M3. The intersection products 2 are Mi.Mj = Li.Lj = −δij, and Mi.Lj = δij. So there is a map π : Sks → Pks , whose exceptional divisors are the Li (with the convention that Li is over pi). The other three exceptional lines Mi 30 ASHER AUEL AND MARCELLO BERNARDARA

2 are the strict transform of the lines in Pks joining pairs of the three points (with the convention that Mi corresponds to the line not going through pi). There is another birational morphism 2 η : Sks → Pks contracting the Mi to three points q1, q2, q3, and sending Li to lines joining two of those three points. We end up with the following diagram

(9.2) Sks η π } ! 2 φ 2 Pks / Pks , where φ is the well-known Cremona involution, a birational self-map of the projective plane of degree 2. This description allows us to present the Picard group of Sks in a way convenient to compare the base change of the semiorthogonal decompositions of Blunk–Sierra–Smith and Karpov– Nogin 3-block decomposition. Indeed, the Picard group of Sks has rank 4 and is generated by the exceptional lines Li and Mi, with the relations Li + Mj = Lj + Mi. If we denote by ∗ H = π O 2 (1), we have that H = L1 + L2 + M3. The anticanonical divisor KS s is then Pks k 0 ∗ KS s = L1 + L2 + L3 + M1 + M2 + M3. On the other hand, if we denote by H = η O 2 (1), k Pks 0 we have H = M1 + M2 + L3 = −KSks − H. To describe the semiorthogonal decomposition (9.1), Blunk, Sierra, and Smith construct s vector bundles over Sk that descend to S in the following way [27]. The ﬁrst one is just OSks . To deﬁne the second vector bundle, consider the following rank 2 vector bundles

J1 = O(L3 + M2) ⊕ O(L2 + M3), (9.3) J2 = O(L1 + M3) ⊕ O(L3 + M1), J3 = O(L1 + M2) ⊕ O(L2 + M1), on Sks . The presentation (9.3) shows that J = J1 ⊕ J2 ⊕ J3 is Galois invariant. Blunk, Sierra, and Smith assert that J descends to a vector bundle J of rank 6 on S and they consider ⊕2 Q = End(J). On Sks , we remark that Ji = O(H − Li) . Thus, by base change, we get that s ⊕2 ⊕2 ⊕2 Q ⊗ k = End(O(H − L1) ⊕ O(H − L2) ⊕ O(H − L3) ), which is Morita equivalent to L3 End i=1 O(H − Li) . To deﬁne the third vector bundle, consider the two rank 3 vector bundles

I1 = O(L1 + M2 + M3) ⊕ O(M1 + L2 + M3) ⊕ O(M1 + M2 + L3), (9.4) I2 = O(L1 + L2 + M3) ⊕ O(L1 + M2 + L3) ⊕ O(M1 + L2 + L3), on Sks . The presentation (9.4) shows that the sum I := I1 ⊕ I2 is Galois invariant. Blunk, Sierra, and Smith assert that I descends to a vector bundle I of rank 6 on S and they consider 0 ⊕3 ⊕3 ⊕3 s B = End(I). On Sk , we remark that I1 = O(H ) = O(−KSks − H) , and I2 = O(H) . s ⊕3 ⊕3 In particular, by base change, we get that B ⊗ k = End(O(H) ⊕ O(−KSks − H) ), which is Morita equivalent to End(O(H) ⊕ O(−KSks − H)). Proof of Proposition 9.1. Let us now recall the construction of the 3 block decomposition over Sks described by Karpov and Nogin [64]. We provide a slightly diﬀerent presentation. Consider 2 the birational morphism π : Sks → Pks , which is the blow up of three points with exceptional b divisors L1, L2 and L3. A semiorthogonal decomposition of D (Sks ) is given by Orlov’s formula (see [84]) as follows: b ∗ b 2 s D (Sk ) = hπ D (Pks ), OL1 , OL2 , OL3 i. 2 Choosing the full exceptional collection {O(−1), O, O(1)} on Pks , we get the semiorthogonal decomposition b s D (Sk ) = hO(−H), O, O(H), OL1 , OL2 , OL3 i. Mutating O(−H) to the left with respect to the whole orthogonal complement we get b s D (Sk ) = hO, O(H), OL1 , OL2 , OL3 , O(−KSks − H)i, DEL PEZZO SURFACES 31

S ind(S) ρ(S) A1 Z ind c2 rk A2 Z ind c2 rk

6.1 S ⊂ RK/kSB(B) 6 1 Q L 2 12 6 B K 3 24 6 6.2 S ⊂ RK/kSB(B) 3 1 L L 1 3 3 B K 3 24 6 6.3 S ⊂ SB(A) × SB(A−1) 3 2 L L 1 3 3 A × A−1 k2 3 24 6 2 6.4 S ⊂ RK/kP 2 1 Q L 2 12 6 K K 1 2 2 2 00 0 0 6.5 S ⊂ RK/kP 2 2 Q × Q k × L 2 12 6 K K 1 2 2 2 0 0 6.6 S ⊂ RK/kP 2 2 k × Q k × L 2 8 5 K K 1 2 2 2 0 00 000 3 6.7 S ⊂ RK/kP 2 3 Q × Q × Q k 2 12 6 K K 1 2 2 2 0 0 3 6.8 S ⊂ RK/kP 2 3 k × Q × Q k 2 8 5 K K 1 2 2 2 6.9 S ⊂ RK/kP 1 1 L L 1 3 3 K K 1 2 2 2 0 0 6.10 S ⊂ RK/kP 1 2 k × L k × L 1 3 3 K K 1 2 2 2 2 2 2 6.11 S ⊂ P × P 1 2 L L 1 3 3 k k 1 2 2 2 3 3 6.12 S ⊂ RK/kP 1 3 k k 1 3 3 K K 1 2 2 2 2 0 0 2 2 6.13 S ⊂ P × P 1 3 k × L k × L 1 3 3 k k 1 2 2 2 2 3 3 2 2 6.14 S ⊂ P × P 1 4 k k 1 3 3 k k 1 2 2 Table 4. The invariants of a del Pezzo surface S of degree 6. Here, the algebras End(V1) = A1 = Q and End(V2) = A2 = B up to Morita equivalence; l1 = Z(A1) = L and l2 = Z(A2) = K are separable cubic and quadratic extensions of k; the columns Z and ind refer to the center and index of Ai; and the columns c2 and rk refer to the 2nd Chern class and rank of Vi. Note that (V1)ks is a direct 0 sum of ⊕(O(H − Li)), while (V2)ks is a direct sum of O(H) ⊕ O(H ). Recall that S is rational if and only if ind(S) = 1, see [43, §2]. See Remark 9.9 for a geometric description of each case.

using [29, Prop. 3.6]. Now we mutate the three exceptional objects OLi to the left with respect to O(H). An easy calculation with the evaluation sequences for Li gives b s (9.5) D (Sk ) = hO, O(H − L1), O(H − L2), O(H − L3), O(H), O(−KSks − H)i. The decomposition (9.5) is a mutation of the 3-block decomposition [64, (3)]. The latter is indeed obtained mutating the three exceptional objects OLi to the right with respect to O(−KSks − H) = O(2H − L1 − L2 − L3), as Karpov and Nogin do. The presentation (9.5) allows the following description of the three blocks:

E = hOi, G = hO(H − L1), O(H − L2), O(H − L3)i, F = hO(H), O(−KSks − H)i. L3 So, the block E corresponds to the ﬁrst component of (9.1). Recall that End i=1 O(H − Li) s is Morita equivalent to Q ⊗ k , and that End(O(H) ⊕ O(−KSks − H)) is Morita equivalent to s B ⊗ k . The claim follows now by Proposition 1.6.

Remark 9.3. A consequence of [25, Thm. 4.1] is that B comes with a natural K/k-unitary involution, equivalently, the corestriction of B from K to k is split. This involution on B was already constructed by Colliot-Thélène,Karpenko, and Merkurjev [41]. Furthermore, the corestriction of Q from L to k is split. Also, B is split by L and Q is split by K. Otherwise, any choices of K and L are possible and any choices of algebras B/K and Q/L are possible, subject to the above restrictions, see [25, Thm. 2.2]. Remark 9.4. Given a del Pezzo surface S of degree 6, Blunk constructs the triple (Q, B, KL) and shows that a toric presentation of S is uniquely determined by the equivalence class under pairwise L-isomorphisms of Q and K-isomorphisms of B, see [25, Thm. 2.4]. On the other hand, a consequence of Blunk’s work is that the isomorphism class of S is uniquely determined by the equivalence class under pairwise k-isomorphisms of Q and B, see [25, Prop. 3.2]. By Theorem 1.17, the semiorthogonal decomposition (9.1) determines Q and B up to pairwise k- linear Morita equivalence, hence k-isomorphism, since the algebras involved are semi-simple of finite rank. We conclude that the semiorthogonal decomposition (9.1) identifies the isomorphism class of S. Now we prove that rationality is equivalent to categorical representability in dimension 0. Proposition 9.5. Let S be a del Pezzo surface S of degree 6. The following are equivalent: (i) S has a k-rational point 32 ASHER AUEL AND MARCELLO BERNARDARA

(ii) S is k-rational (iii) S is categorically representable in dimension 0 (iv) AS is representable in dimension 0 Proof. By [43, §2], S is rational if and only if S(k) 6= ∅ if and only if ind(S) = 1. By [25, Cor. 3.5], S(k) 6= ∅ is equivalent to the triviality of the Brauer classes κ and β and so (9.1) becomes a semiorthogonal decomposition Db(S) = hDb(k), Db(L/k), Db(K/k)i. Hence (i) is equivalent to (ii), which implies (iii). On the other hand, suppose that S has Picard rank 1. If S is categorical representable in dimension 0, then there is a semiorthogonal decomposition b b b b (9.6) D (S) = hD (l1/k), D (l2/k), D (l3/k)i, base-changing to a 3-block exceptional collection over ks . Hence, up to mutation, the decomposition (9.6) equals the decomposition (9.1) and hence AS is representable in dimension 0 and both the Brauer classes of Q and B are trivial, hence S(k) 6= ∅ by [25, Cor. 3.5]. Thus (iii) implies (iv), which implies (i). If the Picard rank of S is > 1, then S is not minimal and we can consider its minimal model S0, which is a del Pezzo surface of degree ≥ 7. We conclude in this case by appealing to the results from the previous sections. Remark 9.6. Colliot-Thélène,Karpenko, and Merkurjev [41, Rem. 4.5] provide a geometric argument to show that the splitting of K implies the non-minimality of S. This can also be seen via the description of B = End(I), where I is the vector bundle constructed above. Indeed, the splitting of K, i.e., B =∼ A × Aop, means that both H⊕3 and 0 (H0)⊕3 descend to vector bundles of rank 3 on S, hence both H and H are Galois invariant line bundles, with Brauer obstruction to descent being the Brauer classes of A and Aop, respectively. Hence (cf. [100, Prop. 8]) we get birational morphisms S → SB(A) and S → SB(A−1) (compare with [41, Rem. 4.5]). As an interesting consequence, the birational Cremona involution φ from −1 diagram (9.2) descends to a k-birational map SB(A) 99K SB(A ). Lemma 9.7. Let S be a del Pezzo surface of degree 6. The index of X divides 6 and there is always a closed point whose degree equals the index. Proof. There is always a closed point of degree 6 on S. Indeed, the 6 points of intersection of the hexagon of exceptional curves defined over ks is Galois invariant, hence defines a point of degree 6. Since the index is the greatest common divisor of the degrees of all closed points, the index of S divides 6. Considering the Galois action on the hexagon of exceptional curves defined over ks , there exists a quadratic extension K/k (resp. cubic extension) such that SK has a triple (resp. pair) of skew exceptional curves, cf. [39, Lemme 1]. Blowing down, we have SK → S0 where S0 is a Severi–Brauer surface defined over K and also SL → S1 where S1 is a del Pezzo of degree s 8 defined over L. In this later case, a lattice computation over k shows that S1 is actually ∼ 2 an involution surface. If we assume that S has index 2, then so does S0, hence S0 = PK . In particular, S(K) 6= ∅ so S has a closed point of degree 2. Similarly, if we assume that S has index 3, then so does S1, hence S1(L) 6= 0 by Springer’s theorem. In particular, S(L) 6= ∅ so S has a closed point of degree 3. Finally, if we assume that S has index 1, then it must have a point of degree relatively prime to 6, hence S(k) 6= ∅ by [43]. We now want to show that the Griffiths–Kuznetsov component is well defined. This also gives a strengthening of [41, Lemma 4.6]. Proposition 9.8. Let S be a del Pezzo surface of degree 6. Then the Griffiths–Kuznetsov component GKS is well defined as a birational invariant as follows. Letting S1 99K S be a birational map, we have: • ind(S) = 6 if and only if both κ and β are nontrivial; there is a semiorthogonal decom- b b position AS1 = hT, D (L/k, κ), D (K/k, β)i DEL PEZZO SURFACES 33

• ind(S) = 3 if and only if κ is trivial and β is nontrivial; there is a semiorthogonal b decomposition AS1 = hT, D (K/k, β)i • ind(S) = 2 if and only if κ is nontrivial and β trivial; there is a semiorthogonal de- b composition AS1 = hT, D (L/k, κ)i where T always denotes a category representable in dimension 0. Proof. We can reduce to the case when S is minimal, equivalently, has Picard rank 1. Indeed, if S is not minimal (and not rational), then there is a birational morphism S → S0 where S0 is either a nonsplit involution surface (when ind(S) = 2) or S0 is a nonsplit Severi–Brauer surface (when ind(S) = 3). We have already treated the Griﬃths–Kuznetsov component in these cases, see §6 and § 7. For the interpretations of κ and β in the nonminimal cases, see Remark 9.9. First, we remark that if β is trivial then S(K) 6= ∅, hence ind(S)|2 by [41, Rem. 4.5]. Similarly, we will argue that if κ is trivial, then S(L) 6= ∅, hence ind(S)|3. Indeed, βL is split by Remark 9.3, so assuming that κ is split implies that SL is categorically representable in dimension 0, hence is rational by Proposition 9.5. As a consequence, if ind(S) = 6, then both β and κ are nontrivial and also S is birationally rigid by Proposition 3.10. Now assume that S has index 3. Then S has a degree 3 point x by Lemma 9.7, and we can appeal to the description of elementary links detailed in §A.3. There is only one elementary 0 0 0 link φ : S 99K S deﬁned by degree three points x on S and x on S . Let X be the del Pezzo surface of degree 3 obtained as a resolution of φ with the following diagram (over ks ): X σ τ ~ φ S / S0

σ0 τ0 2 φ0 2 P / P ∗ Denote by G = τ0 OP2 (1) and Fi be the exceptional divisors of τ0 and by abuse of notations, ∗ ∗ we write L for σ L and G for τ G. Finally, L4,L5,L6 are the exceptional divisors of σ and s F4,F5,F6 the exceptional divisors of τ (over k we are blowing up three points). As calculated in AppendixA, we have that P6 G = 5H − j=1 2Ej P (9.7) Fi = 2H − j6=i+3 Ej for i = 1, 2, 3 P Fi = 2H − j6=i−3 Ej for i = 4, 5, 6. We perform a series of mutations in Db(X) over ks in order to compare End(I), End(J), End(I0), End(J 0) and the residue ﬁelds k(x) and k(x0). Let us choose the following 3-block semiorthogonal decomposition (9.8) b 0 D (S ) = hOS0 |O(G), O(−KS0 − G)|O(−KS0 − G + F1), O(−KS0 − G + F2), O(−KS0 − G + F3)i, which is the original 3-block decomposition of Karpov–Nogin [64]. We Denote E0, F0 and G0 the three blocks in the order given in (9.8) as in Table1. In particular, E0 descends to Db(k), F0 descends to Db(k, B0) and G0 descends to Db(k, Q0). We mutate the second and the third block of (9.8) to the left with respect to OS0 to obtain b 0 (9.9) D (S ) = hO(KS0 + G), O(−G)|O(−G + F1), O(−G + F2), O(−G + F3)|OS0 i, since the top left mutation amounts to tensoring with ωS0 = O(KS0 ). Via the blow-up τ, we get the following four-block decomposition of X: b 0 (9.10) D (X) = hO(KS +G), O(−G)|O(−G+F1), O(−G+F2), O(−G+F3)|OX |OF4 , OF5 , OF6 i. The decomposition (9.10) is made of the four blocks F0, G0, E0, and a new block H0 arising from b 0 0 the blow up, descending to D (k(x )/k). Finally, if we mutate H to the left with respect to OX , 34 ASHER AUEL AND MARCELLO BERNARDARA the evaluation sequence holds: (9.11) b D (X) = hO(KS0 +G), O(−G)|O(−G+F1), O(−G+F2), O(−G+F3)|O(−F4), O(−F5), O(−F6)|OX i.

Now we rewrite all line bundles in terms of H and Li using the relations (9.7). This makes (9.11) into: (9.12) b D (X) = hO(2KX + 2H − L1 − L2 − L3), O(2KX + H)|O(KX + L4), O(KX + L5), O(KX + L6)| O(KX + H − L1), O(KX + H − L2), O(KX + H − L3)|OX i 0 0 0 0 ∨ where the blocks are now F , G , H , and E . We apply the autoequivalence ⊗ωX and mutate the ﬁrst block F0 to the right with respect to its right orthogonal to obtain: (9.13) b ∗ D (X) = hO(L4), O(L5), O(L6)|O(H−L1), O(H−L2), O(H−L3)|O(KX )|O(−σ KS−H), O(H)i, where the blocks are now G0, H0, E0, and F0. Now consider the semiorthogonal decomposition

b (9.14) D (S) = hOS|O(H − L1), O(H − L2), O(H − L3)|O(H), O(−KS − H)i, as in Proposition 9.1. This decomposition has blocks E (descending to Db(k)), G (descending to Db(k, Q)) and F (descending to Db(k, B)) in the order presented in (9.14), as in Table1. This presentation provides, via σ∗, equivalences F ' F0, whence Db(k, B) ' Db(k, B0). On the other hand, G ' H0, whence Db(k, Q) ' Db(k(x0)/k). By symmetry, we have Db(k, Q0) ' Db(k(x)/k). Using Theorem 1.17, we conclude that κ ∈ Br(L) is trivial. If in addition β ∈ Br(K) is trivial, b then S(k) 6= ∅ and S is rational. Otherwise, if β ∈ Br(K) is nontrivial, the category D (K/k, β) is a birational invariant. Indeed, in this case, the index of S is 3 so can have no point of degree 0 2 and all birational maps S 99K S decompose into elementary links of type M6,3. We have proved that ind(S)|3 implies that κ is trivial. Now we handle the case where S has index 2. Then S has a degree 2 point x (cf. Lemma 0 9.7), and we can appeal to the description of elementary links detailed in §A.3. Let φ : S 99K S be the elementary link deﬁned by the two degree two points x on S and x0 on S0. Let X be the del Pezzo surface of degree 4 obtained as a resolution of φ with the following diagram (over ks ):

X σ τ ~ φ S / S0

σ0 τ0 2 φ0 2 P / P ∗ Denote by G = τ0 OP2 (1) and Fi be the exceptional divisors of τ0 and by abuse of notations, ∗ ∗ we write L for σ L and G for τ G. Finally, L4 and L5 are the exceptional divisors of σ and F4 s and F5 the exceptional divisors of τ (over k we are blowing up two points). As in the index 3 case, we perform a series of mutations in Db(X) over ks in order to compare End(I), End(J), End(I0), End(J 0) and the residue ﬁelds k(x) and k(x0). We perform the same ﬁrst mutation as b 0 in the index 3 case, and consider the decomposition (9.9) of D (Sks ). Via the blow-up τ, we get the following four-block decomposition of X:

b 0 (9.15) D (X) = hO(KS + G), O(−G)|O(−G + F1), O(−G + F2), O(−G + F3)|OX |OF4 , OF5 i. The decomposition (9.15) is made of the four blocks F0, G0, E0, and H0. The latter arises from b 0 the blow-up and descends to D (k(x )/k). Finally, if we mutate OF4 and OF5 to the left with respect to OX , the evaluation sequence holds: (9.16) b D (X) = hO(KS0 + G), O(−G)|O(−G + F1), O(−G + F2), O(−G + F3)|O(−F4), O(−F5)|OX i. DEL PEZZO SURFACES 35

Now we rewrite all line bundles in terms of H and the Li using the relations in §A.3 for the link M6,2:

G = 3H − L1 − L2 − L3 − L4 − 2L5

Fi = H − Li − L5, i = 1,..., 4

F5 = 2H − L1 − L2 − L3 − L4 − L5 This makes (9.16) into: (9.17) b D (X) = hO(KX + L4), O(KX + L5)|O(KX + H − L1), O(KX + H − L2), O(KX + H − L3)| O(−H + L4 + L5), O(KX + H)|OX i, 0 0 0 0 where the blocks are F , G , H , and E . Now we mutate the left orthogonal to OX to the right to obtain: b ∗ (9.18) D (X) = hOX |O(L4), O(L5)|O(H − L1), O(H − L2), O(H − L3)|O(−σ KS − H), O(H)i where the blocks are E0, F0, G0, and H0. Now consider the semiorthogonal decomposition b (9.19) D (S) = hOS|O(H − L1), O(H − L2), O(H − L3)|O(H), O(−KS − H)i, as in Proposition 9.1. This decomposition has blocks E (descending to Db(k)), G (descending to Db(k, Q)) and F (descending to Db(k, B)) in the order presented in (9.19), as in Table1. This presentation provides, via σ∗, equivalences E ' E0, and G ' G0, whence Db(k, Q) ' Db(k, Q0). On the other hand, F ' H0, whence Db(k, B) ' Db(k(x0)/k). By symmetry, we have Db(k, B0) ' Db(k(x)/k). Using Theorem 1.17, we conclude that β ∈ Br(K) is trivial. If in addition κ ∈ Br(L) is trivial, then S(k) 6= ∅ and S is rational. Otherwise, if κ ∈ Br(L) is nontrivial, the category Db(L/k, Q) is a birational invariant. Indeed, in this case, the index 0 of S is 2 so can have no point of degree 3 and all birational maps S 99K S decompose into elementary links of type M6,2. We have proved that ind(S)|2 implies that β is trivial. Remark 9.9. We now describe the geometry of the all possible cases listed in Table4. In particular, for the nonminimal cases, we describe how the classes κ and β are related to the Brauer classes arising from their minimal model. Cases 6.1, 6.2, 6.4, and 6.9 are minimal since they have Picard rank 1. Cases 6.9–6.14 are rational by Proposition 9.5. Case 6.3 is the blow up of the Severi–Brauer surface SB(A) in a point x of degree 3 with residue ﬁeld L, via the natural projection of S ⊂ SB(A) × SB(A−1). In fact, S resolves the −1 standard Cremona quadratic transformation SB(A) 99K SB(A ). Recall that B = End(I) = End(I1 ⊕ I2) and notice that I1 and I2 are the pull-backs of the natural rank 3 vector bundles on SB(A) and SB(A−1), respectively. Over ks , the block G is obtained by right mutation 2 of the category generated by the exceptional divisors of Sks → Pks . From this, we see that Db(L/k, κ) ' Db(k(x)/k). Hence κ is trivial and k(x) = L. We now argue that cases 6.5–6.8 are blow ups of an involution variety associated to (A, σ) in 0 2 a point x of order 2 with residue ﬁeld K, where the center of C0 is isomorphic to L (or k ). The minimal model π : S → S0 has index 2 and degree > 6, hence must be an involution surface of index 2. Over ks , consider the diagram: S

π S1

σ τ Ñ y 1 1 2 S0 = P × P P where σ blows up a point q with exceptional divisor F , τ blows up two points p1 and p2 with exceptional divisors L1 and L2, and η blows up a point p3 with exceptional divisor L3. Here, only π is deﬁned over k and the exceptional divisor is L3 + F . Let us denote by H1 and H2 36 ASHER AUEL AND MARCELLO BERNARDARA

1 1 2 the two ruling of P × P and by H the hyperplane section of P , and by abuse of notations, all their pullbacks. Then we have Hi = H − Li for i = 1, 2 and F = H − L1 − L2. Now we consider the 3-block decomposition b 1 1 D (P × P ) = hO(−H1 − H2)|O|O(H1), O(H2)i b b where the first block descends to D (k, A) and the third block descends to D (k, C0). Via the blow up π, we obtain b D (S) = hO(−L3), O(−F )|O(−H1 − H2)|O|O(H1), O(H2)i where the first block, call it H, descends to Db(k(x)/k). Mutating this block to the right with respect to O(−H1 − H2) and mutating O(−H1 − H2) to the right with respect to its right orthogonal, and substituting the previous relations, we obtain: b D (S) = hH|O|O(H − L1), O(H − L2)|O(H − L3)i so that the last two blocks descend together to Db(L/k, κ). We conclude that Db(L/k, κ) = b b b D (k, C0) × D (k, A). It also follows that H descends to D (K/k, β) and we conclude that β is trivial and K = k(x). By comparing with the index 2 cases on Table3, we see that: case 6.5 is the blow up of case 0 00 0 0 8.3, Q is Morita equivalent to C0(A, σ) and also Q is the corestriction of Q from L /k and is Morita equivalent to A; case 6.6 is the blow up of case 8.4 (a quadric of Picard rank 1) and 0 0 00 000 Q is Morita equivalent to C0(A, σ); case 6.7 is the blow up of case 8.5 and Q ⊗ Q ⊗ Q is trivial; and case 6.8 is the blow up of either cases 8.6 or 8.7 (which are anyway birational to each other), in fact, it is the resolution of this birational map. Finally, when S is rational, we can see that: case 6.10 is the blow up of a rational quadric of 2 Picard rank 1 along a point of degree 2 or and is not the blow up of P ; case 6.11 is the blow 2 up of P in a point of degree 3 with residue field L and cannot be the blow up of a quadric; 1 1 case 6.12 is the blow up of P × P in a point of degree 2 or the blow up of a rational quadric of Picard rank 1 in two rational points (this resolves the rational map between these two); case 1 1 0 2 6.13 is the blow up of P × P in a point of degree 2 with residue field L or of P in a union of a rational point and a point of degree 2 with residue field L0 (this resolves a birational map between the quadric and the Hirzebruch surface of degree 1). Case 6.14 is totally split.

min min Remark 9.10. Let V1 = J and V2 = I (recall Definition 2.10). These vector bundles are tilting bundles for the blocks F and G, respectively, and have the following properties: A1 = End(V1) is Morita equivalent to Q and A2 = End(V2) is Morita equivalent to B; V1 is indecomposable if and only if L is a field; and V2 is indecomposable if and only if K is a field. In particular, both Vi are indecomposable if and only if S is minimal. We list the ranks and 2nd Chern classes of the vector bundles Vi on Table4. The calculation of the second Chern classes of the vector bundles V1 and V2 is easily obtained s by their description over k . In particular, we notice that ind(S) = gcd(c2(V1), c2(V2)) unless ind(S) = 6, in which case gcd(c2(V1), c2(V2)) = 12. When ind(S) = 6, we have to appeal to a ⊕2 particular generator (ωS works) of the remaining block to obtain a bundle with second Chern class equal to 6.

10. del Pezzo surfaces of degree 5 Let S be a Del Pezzo surface of degree 5. It is a classical fact (announced by Enriques [48] and proved ﬁrst by Swinnerton-Dyer [99]), that S(k) 6= ∅ (see [95] for a diﬀerent proof). 2 The base change Sks is the blow-up of Pks at four points in general position. Such a surface has 10 exceptional lines. Fix x ∈ S(k). If x lies in the intersection of two exceptional lines, then S is not minimal (see [82, 29.4.4.(v)]). So we can suppose that x does not lie on the intersection of two exceptional lines and consider the geometric construction described by Manin to show that S is rational: let X → S be the blow-up of x, and D its exceptional divisor. Then there are 5 pairwise nonintersecting exceptional lines L1,...,L5, where L5 = D, on Xks . DEL PEZZO SURFACES 37

Manin shows that on X there is an exceptional divisor Z ⊂ X whose contraction gives a birational map onto a del Pezzo surface of degree 9. Since the target also has a rational point, 2 we have a birational map π : X → Pk. We have a diagram of birational morphisms X π ε

2 Pk S,

2 where π : X → Pk is the blow-up of a closed point of degree 5 in general position. Passing to the algebraic closure, we can describe these birational maps in the following way: 2 let p1, . . . , p5 be five points in general position on Pks . Then Xks is a del Pezzo of degree four which has 16 exceptional lines: five of them are the exceptional divisors E1,...,E5 of π, ten of them are the strict transforms of the lines Li,j passing through the points pi and pj. The last one is the strict transform D of the conic through the five points pi. This is, indeed, the exceptional divisor of the blow-up ε : Xks → Sks . On the other hand, the lines E1,...,E5 all meet twice D, and it can be checked that they are the only exceptional lines on Xks with such property. Since D is defined over k; it follows that E1,...,E5 are Galois-invariant and hence the cycle Z is the descent of the disjoint union of the Ei’s. Conversely, given any point of degree 2 5 (geometrically) in general position on Pk, we can blow it up, and then blow down the strict transform of the conic through the point to obtain a del Pezzo surface of degree 5. Moreover, the surface Sks is a del Pezzo surface of degree 5 and can therefore be realized 2 as the blow up of Pks in four points in general positions. Given Sks , this can be realized by the choice of 4 pairwise nonintersecting exceptional lines L1,...,L4. It is easy to see, via the previous construction, that we have five choices, one for each of the points pi. Once we fix such a point, it is then enough to consider all the lines joining it to the other four points, which are blown up by π to exceptional lines (call them loosely L1,...,L4), which are, in turn, blown down by ε to 4 exceptional nonintersecting lines. Let us then fix p5, so that Li is the (strict 2 transform of) the line through p5 and pi, and consider the blow-down η : Sks → Pks . This latter map is not, in general, defined over k, so we will avoid the “overline” notation to mark this difference. We end up with the following diagram:

_ (10.1) {E1,...,E5} / Xks o ? {D,L1,...,L4} ε

π & _ Sks o ? {x, L1,...,L4}

η Ò 2 φ 2 _ {p1, . . . , p5} / Pks / Pks o ? {x, q1, . . . , q4}, where qi are the points blown-up by η, and φ the birational map obtained by composition. ∗ Let us denote by OS s (H) = η O 2 (1). We can assume that this line bundle is not deﬁned k Pks 2 over k, since we can suppose that S is minimal. Otherwise, S is the blow-up of Pk. We will explicitly use this construction to show that the three-block exceptional collection described by Karpov and Nogin [64] over ks descends to a zero-dimensional semiorthogonal decomposition of Db(S).

Proposition 10.1. Any del Pezzo surface of degree 5 is k-rational and is categorically representable in dimension 0. In particular, there is a degree 5 ´etale k-algebra l and a semiorthogonal decomposition b b AS = hD (k), D (l/k)i hence AS is also representable in dimension 0. Moreover, l is a ﬁeld if and only if ρ(S) = 1. 38 ASHER AUEL AND MARCELLO BERNARDARA

S ind(S) A1 c2 rk A2 c2 rk 5 S ⊂ Pk 1 k 2 2 l 20 5 Table 5. The invariants of a del Pezzo surface S of degree 5 of Picard rank 1. Here, the algebras End(V1) = k and End(V2) = l are listed up to Morita equivalence; l is an ´etale k-algebra of degree 5, and is a ﬁeld if and only if ρ(S) = 1; and Z, per, and ind refer to the center, period, and index of Ai; and c2 and rk refer to the 2nd Chern class and rank of Vi. Note that (V1)ks is the unique extension of O(H) by O(KS − H), while (V2)ks is the direct sum L4 O(H) ⊕ i=1 O(Li − KS − H).

Proof. Over ks , Karpov and Nogin [64, §4] provide the following 3-block decomposition: b s (10.2) D (Sk ) = hOSks | F | OSks (H), OSks (L1 − KSks − H),..., OSks (L4 − KSks − H)i, where F is the rank 2 vector bundle given by the nontrivial extension

(10.3) 0 −→ OSks (−KSks − H) −→ F −→ OSks (H) −→ 0.

These blocks are denoted E, F, and G. Consider the rank 5 vector bundle on Sks 4 M (10.4) V = OSks (Li − KSks − H) ⊕ OSks (H), i=1 and BV = EndSks (V ). Since the five line bundles form an exceptional block, we have that s 5 BV ' (k ) and V is a tilting bundle for the block G. We are going to show that V descends to k, hence BV descends to a degree 5 extension l of k, and G descends to k. To this end, ∗ b b ∗ b b recall that the functors ε : D (S) → D (X) and ε : D (Sks ) → D (Xks ) are fully faithful. We b analyze the pull-back of the three-block collection (10.2) as an exceptional collection in D (Xks ) to deduce the descent. In order to do that, we first stress the structure of the Picard group of Xks . ∗ ∗ ∗ On one hand, we have the line bundle OX s (H) = ε OS s (H) = ε (η O 2 (1)). We have the k k Pks five exceptional lines D (the exceptional locus of ε) and L1,...,L4 (by abuse of notation, we ∗ ∗ denote Li = ε Li). We have that KXks = ε KSks − D. ∗ On the other hand, we have the line bundle OX s (G) = π O 2 (1), and the five exceptional k Pks P5 lines E1,...,E5 of the blow-up π. We have that KXks = −3G + i=1 Ei. The divisor D is the strict transform via π of the conic passing through the five points pi. P5 Hence D = 2G − i=1 Ei, so that −KXks = D + G. For each 1 ≤ i ≤ 4, the divisor Li is the strict transform of the line through p5 and pi, so that Li = G − E5 − Ei. Finally, the birational map φ is given by the system of cubics through the five points pi which have multiplicity 2 in p5. Indeed, the map φ is not defined over the curve given by the conic D and the four lines joining pi to p5. This is a curve C of degree 6. The map φ can be written as three homogeneous polynomials of the same degree d, which we proceed to determine. The degeneracy locus C of φ is then the zero locus of the determinant of the Jacobian matrix. This is a 3 × 3 matrix with entries of degree d − 1, hence the polynomial defining C has degree 3(d − 1). Thus 6 = deg(C) = 3(d − 1), from which we deduce that d = 3. This implies that φ is given 2 by a linear system of cubics. The hyperplane section H of the target Pks corresponds then to Pn the linear system 3G − i=1 mixi, where xi are the points of multiplicity mi > 0 of the map φ. By construction, it is clear that the points of positive multiplicity are exactly the pi (they are P5 transformed via φ into lines), so we get the linear system 3G − i=1 mipi. This linear system P5 2 must have degree 1, so we get 9 − i=1 mi = 1. Since φ is given by the cubics passing through the points pi, with multiplicity two in p5, we deduce that m5 = 2 and mi = 1 for 1 ≤ i ≤ 4. From this, we get that 4 5 ∗ X X (10.5) ε H = 3G − 2E5 − Ei = 3G − Ei − E5 = −KXks − E5. i=1 i=1 DEL PEZZO SURFACES 39

On the other hand, consider, for any 1 ≤ i ≤ 4, the divisor Li − KSks − H and pull it back ∗ s via ε. Recall that ε KSks = KXks + D, and that Li = H − Ei − E5 over Xk . We then get ∗ ε (Li − KSks − H) = G − E5 − Ei − KXks + D − H. using equation (10.5), we substitute L to get: ∗ (10.6) ε (Li − KSks − H) = G + D − Ei = −KXks − Ei. Using equations (10.5) and (10.6), the block G pulls back via ε to the exceptional block

(10.7) hOXks (−KXks − E1),... OXks (−KXks − E5)i, b in D (Xks ), and where we have performed a mutation of the completely orthogonal bundles in the block to arrive at this ordering. It follows that 5 5 M M ε∗V = (−K − E ) = ω∨ ⊗ (−E ), OXks Xks i Xks OXks i i=1 i=1 hence ε∗V descends to a vector bundle of rank 5 on X and V descends to a vector bundle (again denoted by V ) of rank 5 on S since ε∗ is fully faithful. We see that End(V ) is then isomorphic 2 to the structure sheaf l of the degree 5 point in Pk that is blown up by π, which is an k-étale algebra of degree 5. We conclude that G descends to a block over k equivalent to Db(l/k). It is now sufficient to prove that F descends to k, which would imply that F ' Db(k). Since E and G descend to blocks of Db(S) defined over k, so does F, being the orthogonal complement. Hence by Theorem 2.11, F descends to a block equivalent to Db(k, α) for some α ∈ Br(k). We proceed to show that α is trivial. To this end, consider the semiorthogonal decomposition (10.2). Orlov’s formula applied to the blow up ε gives the following 4-block semiorthogonal decomposition: b ∗ s (10.8) D (Xk ) = hOXks (−D)|OXks |ε F |OXks (−KXks − E1),..., OXks (−KXks − E5)i where we used the identifications (10.5) and (10.6) in writing G. Mutating G to the left with respect to its left orthogonal, we obtain: b ∗ s (10.9) D (Xk ) = hOXks (−E1),..., OXks (−E5)|OXks (−D)|OXks |ε F i. As the first block, the mutation of G, is generated by the exceptional divisors of the blow up ∗ π, by Orlov’s blow up formula, it follows that hOXks (−D)|OXks |ε F i can be identified with ∗ b 2 π D (Pks ). Since π, as well as the line bundles OX (−D) and OX , is defined over k, we arrive ∗ b 2 b at a 3-block decomposition π D (Pk) = hOX (−D), OX , D (k, α)i. By the uniqueness of 3-block 2 decompositions on P (see [52] or Proposition 4.1), and by Corollary 1.18, we conclude that α is ∗ ∗ trivial. Moreover, ε F can be mutated into an exceptional line bundle π O 2 (i) via a sequence Pks ∗ b 2 ∗ of mutations inside π D (Pks ), which are a posteriori, all defined over k. It follows that ε F ∗ can be mutated to π OP2 (i), hence F descends to a k-exceptional vector bundle of rank 2.

Remark 10.2. Let V1 = F and V2 = V . These vector bundles are tilting bundles for the blocks F and G, respectively, and have the following properties: A1 = End(V1) is Morita equivalent to k and A2 = End(V2) is Morita equivalent to l; V1 is always indecomposable; and V2 is indecomposable if and only if l is a ﬁeld. In particular, both Vi are indecomposable if and only if S is minimal. We list the ranks and 2nd Chern classes of the vector bundles Vi on Table5. The calculation of the second Chern classes of the vector bundles V1 and V2 is easily obtained s ⊕2 by their description over k . In particular, we notice that 1 = ind(S) = gcd(c2(V1), c2(V2), c2(ωS )) showing that we must include a generator of each block.

Part 3. Appendix: Explicit calculations with elementary links Appendix A. Elementary links for non-rational minimal del Pezzo surfaces In this appendix, we consider all possible links between two non-rational minimal del Pezzo surfaces S and S0. Let us brieﬂy sketch the notion of elementary link in Sarkisov’s program from [58]. We consider π : S → Z to be a minimal geometrically rational surface with an extremal 40 ASHER AUEL AND MARCELLO BERNARDARA

contraction. Hence one obtains that either Z is a point and S a minimal surface with Picard number 1, or Z is a Severi–Brauer curve and S → Z is a minimal conic bundle, and the Picard number of S is 2. If S → Z and S0 → Z0 are such extremal contractions, an elementary link is a birational map 0 φ : S 99K S of one of the following types: Type I) There is a commutative diagram

σ S o S0

ψ ZZo 0 where σ : S0 → S is a Mori divisorial elementary contraction and ψ : Z0 → Z is a morphism. In this case, Z = Spec(k), ρ(S) = 1, S is a minimal del Pezzo, and S0 → Z0 is a conic bundle over a Severi–Brauer curve. type II) There is a commutative diagram

σ τ S o X / S0

=∼ ZZo 0 where σ : X → S and τ : X → S0 are Mori divisorial elementary contractions. In this case, S and S0 have the same Picard number. type III) There is a commutative diagram

σ S / S0

ψ Z / Z0 where σ : S → S0 is a Mori divisorial elementary contraction and ψ : Z → Z0 is a morphism. These links are inverse to links of type I. type IV) There is a commutative diagram

=∼ S / S0

Z Z0 ψ ψ0

# { Spec(k) where Z and Z0 are Severi–Brauer curves and ψ and ψ0 are the structural morphisms. This link amounts to a change of conic bundle structure on S. 0 Iskovskikh shows that any birational map S 99K S between minimal geometrically rational surfaces factors into a finite sequence of elementary links [58]. We are interested in the case where S and S0 are both non k-rational del Pezzo surfaces of Picard rank 1. Thanks to Iskovskikh’s classification, a link of type I (resp. III) can happen in the non k- rational cases only if S (resp. S0) has either degree 8 and a point of degree 2, or has degree 4 and a rational point [58, Thm. 2.6]. It follows that if we assume S to not be of this type, then we only have to consider links of type II where ρ(S) = ρ(S0) = 1. By Iskovskikh’s classification, there is a finite list of such links. In particular, if we assume S to not be k-rational, and S0 not isomorphic to S, then we have that deg(S) = deg(S0) can be only 6, 8 or 9, and we are left with five possible links. DEL PEZZO SURFACES 41

deg(S) deg(x) Transformation Matrix 2 1 3 M = 9 9,3 −3 −2 5 4 6 M = 9,6 −6 −5 3 2 8 4 M = 8,4 −4 −3 2 1 2 M = 6 6,2 −3 −2 3 2 3 M = 6,3 −4 −3 Table 6. The possible links of type II between nonisomorphic non-k-rational Del Pezzo surfaces. The transformation matrix expresses the base change from ∗ ∗ σ ωS,E to τ ωS0 ,F .

0 Let φ : S 99K S be a link of type II between non-k-rational non-isomorphic surfaces, and recall that we assume that if S has degree 8 (resp. 4), there is no degree 2 (resp. rational) point on S. Then deg(S) = deg(S0) and there is a closed point x in S of degree d such that φ is resolved as X σ τ SS0 where σ is the blow-up of x and τ is the blow up of a point x0 of degree d on S0. Let E be the exceptional divisor of σ and F be the exceptional divisor of τ. If one considers the Z-bases ∗ ∗ (σ ωS,E) and (τ ωS0 ,F ) of Pic(X), the birational map φ can be described by the transformation matrix between these two bases. We list all possibilities from [58, Thm. 2.6] in Table6. In order to understand the behavior of the semiorthogonal decompositions of S and S0 under birational maps, it is enough to consider the links listed in Table6. We will proceed as follows: 0 0 given a link φ : S 99K S , we describe the birational map φ : Sks 99K Sks . Notice that φ is not a link, since over ks , we can factor σ into a finite sequence of blow-ups (actually, deg(x) of them). In order to describe φ we will consider the description of the Picard group of Sks . If deg(S) = 2 9, then φ is described by a linear system on Pks , the so-called homaloidal system of φ. If 3 deg(S) = 8, we find similarly a homaloidal system on the quadric Sks ⊂ Pks . Finally, if 2 0 2 deg(S) = 6, we have to choose models Sks → Pks and Sks → Pks , and describe how φ corresponds 2 to a homaloidal system on Pks . 2 Pr In general, let us consider a linear system on P of the form G = nH − i=1 mipi, where 2 n > 0 and mi > 0 are integers, H denotes the hyperplane divisor, and pi are points on P . Such 2 2 a linear system defines a birational map P 99K P if and only if deg(G) = 1 and the curves in the linear system are rational. We can resolve the birational map by blowing up the points pi, and call X the blow up. We obtain then the following conditions on n and mi: Pr 3n − 3 = i=1 mi since G.KX = 3 (A.1) 2 Pr 2 2 n − 1 = i=1 mi since G = 1 In order to describe the system, we will extensively use the Cauchy–Schwartz inequality r r 2 X X 2 mi ≤ r mi i=1 i=1 to bound the possible values of n. In particular, we obtain that 9(n − 1) ≤ r(n + 1). Moreover, it is clear that if n = 1, then G = H defines an isomorphism. Hence we have that r ≥ 3. Let us spell out all the possible birational transforms with 3 ≤ r ≤ 6. 42 ASHER AUEL AND MARCELLO BERNARDARA

If r = 3, then n = 2. Conditions (A.1) give mi = 1. These are the standard quadratic 2 2 transformations φ2 : P 99K P . P4 2 If r = 4, then n = 2. Conditions (A.1) give i=1 mi = 3, which is impossible since we only consider mi > 0. It follows in particular that n = 2 if and only if r = 3. If r = 5, then n = 3. Conditions (A.1) give m1 = ... = m4 = 1 and m5 = 2, so that there is only one possibility (up to renumbering the points). We denote these birational maps by 2 2 φ3 : P 99K P . If r = 6, then n ≤ 5. Conditions (A.1) give two possibilities. The ﬁrst one is n = 5, 2 2 mi = 2. We denote these birational maps by φ5 : P 99K P . The second possibility is n = 4, m1 = m2 = m3 = 1 and m4 = m5 = m6 = 2 (up to renumbering the points). One can check 2 2 that the birational map φ4 : P 99K P is the composition of two standard quadratic transforms, 2 2 the ﬁrst one φ2 : P 99K P blows up p1, p2, and p3, so that p4, p5, and p6 belong to the target 2 P . The second standard quadratic transform blows-up p4, p5, and p6. 2 With this calculation in mind, we are able to describe the homaloidal systems of Pks for the links in degree 6, 8, and 9 del Pezzo surfaces in Table6.

2 A.1. Degree 9. If deg(S) = 9, then Sks ' Pks . Hence we are considering a birational map 2 2 φ : Pks → Pks .

The link M9,3 base changes to the following diagram (we omit the overlines for ease of notations): X σ τ

2 ~ φ 2 P / P where σ blows up three points p1, p2, and p3. Hence φ is the standard quadratic transformation. ∗ ∗ Since G = 2H − L1 − L2 − L3, we have that σ O(−1) = τ O(1) ⊗ ωX .

The link M9,6 base changes to the following diagram (we omit the overlines for ease of notations): X σ τ

2 ~ φ 2 P / P where σ blows up six points p1 . . . , p6. As explained above, we have two possibilities: φ is either of type φ5 or φ4. Checking the action of the matrix M9,6 one gets that φ is of type φ5, since all mi must be equal. Since G = 5H − 2L1 − 2L2 − 2L3 − 2L4 − 2L5 − 2L6, we have that ∗ ∗ ⊗2 σ O(−1) = τ O(1) ⊗ ωX . These considerations lead to a simple proof of Amitsur’s theorem in the case of degree 3 central simple algebras.

+ Proposition A.1. Let S be a non-k-rational minimal surface of degree 9, and let TS (resp. − TS ) be the category generated by the descent of a hyperplane section OSks (1) (resp. OSks (−1)). 0 0 For any birational map φ : S 99K S to a minimal surface S (which must be of degree 9), we + + + − have either an equivalence TS ' TS0 , or an equivalence TS ' TS0 .

0 + Proof. It is easy to see that any elementary link S 99K S gives an equivalence between TS and − ⊗2 TS0 , just by pull-back to X and tensor by either ωX or ωX .

3 A.2. Degree 8. If deg(S) = 8 and S is an involution surface, then Sks ⊂ Pks is a quadric surface. We are interested in the hyperplane section (1) and in (2) = ω∨ , the anticanonical O O Sks divisor. The latter is always deﬁned on S. DEL PEZZO SURFACES 43

The link M8,4 base changes to the following diagram (we omit the overlines for ease of notations): X σ τ φ S / S0 where σ blows up four points p1, . . . , p4. Using the action of the matrix on the Picard group of ∗ ∗ ⊗2 X, we get that σ O(1) = τ O(1) ⊗ ωX .

As a corollary, we can see that involution surfaces with Picard rank one and no point of degree ≤ 2 are birationally semirigid. We can give a further refinement of this result purely using the algebraic theory of quadratic forms. Recall that an involution surface has Picard rank 2 or 1 depending on whether it has trivial or nontrivial discriminant, respectively, see Example 3.3. Proposition A.2. Let S and S0 be k-birational involution surfaces over an arbitrary field k. Then S and S0 are k-isomorphic in the following cases: 0 3 (i) S and S are anisotropic quadrics in P , (ii) S (and hence S0) has nontrivial discriminant and no rational point, (iii) S (and hence S0) has index 4. Proof. We know that S is rational as soon as it has a rational point, in which case it can have k-birational yet nonisomorphic partners. Hence we can assume that S(k) = ∅. By considering the Galois action on the the Picard groups, we see that k-birational involution surfaces have the same discriminant. Now let S and S0 be associated to quadratic pairs (A, σ) and (A0, σ0). Part (i) is a classical result in the theory of quadratic forms, see [78, XII.2.2]. This handles the case when both A and A0 are split, hence we may assume that A is not split. We recall a result proved by Arason for quadric surfaces and generalized by Tao [104, Thm. 4.8(b)] to involution surfaces: ( hAi if the discriminant is nontrivial (A.2) ker(Br(k) → Br(k(S))) = + − hC0 ,C0 i if the discriminant is trivial Any k-birational isomorphism between S and S0 induces a k-isomorphism of function fields k(S) =∼ k(S0), under which the Brauer group kernels ker(Br(k) → Br(k(S))) and ker(Br(k) → Br(k(S0))) coincide. We now proceed according to cases. For part (ii), S and S0 have nontrivial discriminant, so (A.2) implies that the cyclic subgroups of the Brauer group generated by A and A0 are the same. However, since both algebras carry involutions of the first kind, they are of period 2 in the Brauer group (we are assuming they are not split). Thus A and A0 are Brauer equivalent, hence are k-isomorphic since they have the same degree. We will now show that the quadratic pairs (A, σ) and (A0, σ0) become adjoint ∼ 0 0 to anisotropic quadratic forms over F = k(SB(A)) = k(SB(A )) (and SF and SF are still F - birational), which by case (i) implies that they are isomorphic over F , which in turn, by the following Lemma A.3, implies that they are isomorphic over k. This anisotropicity statement follows, in the case when S (hence S0) has index 4, i.e., that A =∼ A0 is division, from Tao [104, Prop. 4.18, Cor. 4.20, Cor. 4.21] (see also Karpenko [61, Thm. 5.3] in greater generality). In the case when S (hence S0) has index 2, which under our assumptions implies that A =∼ A0 has index 2, a result of Karpenko [62, Thm. 3.3], stating that the Witt index of a quadratic pair (A, σ) over F is divisible by the index of the A, implies that the quadratic pairs σ and σ0 are either anisotropic or hyperbolic over F (see also [16]). The later is impossible, since by assumption the quadratic pairs have nontrivial discriminant, which remains nontrivial over F . Another way to see the index 2 case, at least when the characteristic is not 2, is by writing A = M2(H), where H is a quaternion algebra, and interpreting the involution (A, σ) as a (−1)-hermitian form of rank 2 over (H, τ), where τ is the standard involution, and similarly for (A0, σ0), then invoking the result of Parimala, Sridharan, and Suresh [89] that the involutions remain anisotropic over k(SB(H)), hence over F , since F/k(SB(H)) is purely transcendental. 44 ASHER AUEL AND MARCELLO BERNARDARA

To finish part (iii), we need only deal with the case of index 4 and trivial discriminant, in which case (A.2) implies an equality of two Klein four subgroups of the Brauer group. By the fundamental relations for Clifford algebras [67, Thm. 9.14], we have the equality of Brauer + − ± ± classes [A] = [C0 ] + [C0 ], and we can rule out [A] = [C0 ] since ind(A) = 4 while ind(C0 ) ≤ 2 ± 0 0 (in fact, we see that ind(C0 ) = 2), and similarly for A . We deduce that A and A are each the unique element of index 4 in their respective Klein four Brauer group kernels. Hence A and A0 are Brauer equivalent, thus are k-isomorphic since they have the same degree. Also, + − 0 the unordered pairs of Clifford algebra components C0 and C0 , associated to A and A , are isomorphic. Thus, by the classification of quadratic pairs of degree 4 and trivial discriminant 0 0 + + − − ± [67, §15.B], both (A, σ) and (A , σ ) are isomorphic to (C0 , τ0 ) ⊗ (C0 , τ0 ), where τ0 is the ± standard involution on the quaternion algebra C0 .

The following result, in characteristic 6= 2, can be seen as a consequence of general hyperbolicity results for orthogonal involutions due to Karpenko [63]. The following direct argument in the case of degree 4 algebras, using the results of [67, §15.B], was communicated to us by Anne Qu´eguiner-Mathieuand works over any ﬁeld.

Lemma A.3. Let σ1 and σ2 be quadratic pairs on a central simple algebra A of degree 4 over a ﬁeld k and let F = k(SB(A)). If σ1 and σ2 become isomorphic quadratic pairs over F then they are isomorphic over k.

Proof. Since (A, σ1) and (A, σ2) become isomorphic over F , their discriminants coincide over 1 1 F , hence coincide over k, since the map H (k, Z/2Z) → H (F, Z/2Z) is injective. Let K/k be the discriminant extension. By the low dimension classiﬁcation of algebras of degree 4 with quadratic pair [67, §15.B], we have that (A, σi) = NK/k(Hi, τi), where τi is the standard involution on the quaternion algebra Hi over K. ι ι Over K, we get (A, σi)K = (Hi, τi) ⊗ ( Hi, τi), where ι is the the non trivial automorphism ι of K/k. Let KF be the compositum of K and F . Over KF , we have that Hi and Hi are ∼ isomorphic, hence A = End(Hi) and σi is adjoint to the quadratic norm form of Hi. Therefore, if the quadratic pairs σi are isomorphic over F , then the norm forms of Hi over KF are isomorphic. In particular, H1 ⊗ H2 is split over KF . Hence, either H1 ⊗ H2 is already split over K, or, by ι ι Amitsur’s theorem, H1 ⊗ H2 = H1 ⊗ H1. Thus H2 is either isomorphic to H1 or H1. In both cases, we get an isomorphism of quadratic pairs σ1 and σ2.

In the remaining cases, we will classify all possible nonisomorphic birational involution surface partners in Proposition C.3.

A.3. Degree 6. Let S be a degree 6 non-k-rational del Pezzo surface. Then, as recalled in Table6, there are two possible types of elementary links. Consider Sks , and recall that there are six exceptional lines, coming into two triples of non-intersecting lines. Each of these triples 2 2 gives a map Sks → Pks , so that we have two ways of blowing down Sks to Pks . The previous considerations show that Sks can be seen as the resolution of a standard quadratic transforma- 0 2 tion. Let H and H denote the pull back of the generic line from each of the P , and let {Li} 0 and {Li} be the two sets of exceptional divisors. In particular, we have two Z-bases for Pic(Sks ), one given by H and the Li, and the other 0 0 0 0 given by H and the Li. We have H = 2H −L1 −L2 −E, and Li = H −Lj −Lk for i 6= j 6= k 6= i. 0 If φ : S 99K S is an elementary link, we suppose that we have chosen the triple Li (and hence 0 2 Li) to describe φ as coming from a homaloidal system on Pks . DEL PEZZO SURFACES 45

The link M6,2 base changes to the following diagram (we omit the overlines for ease of notations): X σ τ ~ φ S / S

σ0 τ0 2 φ0 2 P / P where σ0 blows up three points p1, p2, and p3, and σ blows up two points p4 and p5. Hence φ0 is resolved by blowing up 5 points. In this case, we should calculate which one of the five ∗ ∗ points has coefficient 2 in the homaloidal system of φ0. Let us then denote H = σ σ0O(1), ∗ ∗ G = τ τ0 O(1), Li the exceptional divisor over pi, and Fi the exceptional divisor over qi. ∗ ∗ The matrix M6,2 in Table6 is the transformation matrix from σ ωS,E to τ ωS0 ,F . Since ωS = −3H + L1 + L2 + L3 and ωS0 = −3G + F1 + F2 + F3, we get the following conditions: 3G − F − F − F = 6H − 2L − 2L − 2L − 3L − 3L (A.3) 1 2 3 1 2 3 4 5 F4 + F5 = 3H − L1 − L2 − L3 − 2L4 − 2L5. Since X is a del Pezzo of degree 4, there are only 16 exceptional lines on X, which can be described, in the base H,Li as follows: • The five exceptional lines Li, • The ten strict transforms Li,j of the lines through two of the pi’s. They are of the form H − Lj − Li for any i 6= j. Pr • The strict transform D of the conic through the pi’s. It is of the form 2H − i=1 Li. Using the above description (and the fact that Fi is not of type Lj), it is easy to check that (up to switching 4 and 5) we get that F5 = D and Fi = Li,5 for i 6= 5. Hence the homaloidal system of φ0 is 3H − L1 − ... − L4 − 2L5, which is indeed the only linear system with 5 base points. Our calculation aims at finding the point with coefficient −2. Notice that while we could have switched 4 and 5, the coefficients of L1, L2, and L3 must be 1.

The link M6,3 base changes to the following diagram (we omit the overlines for ease of notations): X σ τ ~ φ S / S

σ0 τ0 2 φ0 2 P / P where σ0 blows-up three points p1, p2 and p3, and σ blows-up three points p4, p5 and p6. Hence φ0 is resolved by blowing-up 6 points. In this case we have two possibilities for the homaloidal system of φ0, and we appeal to the form of the matrix M6,3 to understand which one we are ∗ ∗ ∗ ∗ indeed considering. Let us then denote by H := σ σ0O(1), and G := τ τ0 O(1), by Li the exceptional divisor over pi, and by Fi the exceptional divisor over qi. ∗ ∗ The matrix M6,3 in Table (6) is the transformation matrix from (σ ωS,E) to (τ ωS0 ,F ). Since ωS = −3H + L1 + L2 + L3 and ωS0 = −3G + F1 + F2 + F3, we get the following conditions: 3G − F − F − F = 9H − 3L − 3L − 3L − 4L − 4L − 4L (A.4) 1 2 3 1 2 3 4 5 6 F4 + F5 + F6 = 6H − 2L1 − 2L2 − 2L3 − 3L4 − 3L5 − 3L6. Since X is a del Pezzo of degree 3, there are only 27 exceptional lines on X, which can be described, in the basis (H,Li), as follows: • The six exceptional lines Li. • The ﬁfteen strict transforms Li,j of the lines through two of the pi’s. They are of the form H − Lj − Li for any i 6= j. 46 ASHER AUEL AND MARCELLO BERNARDARA

• The six strict transforms Dj of the conic through the pi’s for i 6= j. It is of the form P 2H − i6=j Li.

Now we use that φ0 is a birational map resolved by a cubic surface, we have hence two possibilities to write G in the basis H, Li. The ﬁrst one is G = 4H−2L1−2L2−2L3−L4−L5−L6, in which case φ0 is the map we called φ4 above, and we observed that this map is the composition of two standard quadratic tranformations. It is easy to check that in this case S would not be minimal, because it would admit a birational morphism onto a Severi–Brauer surfce. P6 We are left with the case where G = 5H − i=1 2Li, in which case φ0 is the map we called φ5 above. Using the above description of the exceptional lines on the cubic and the action of the matrix M3,6 (and the fact that Fi is not of type Lj), it is easy to check that (up to internal permutations of 1, 2, 3 and of 4, 5, 6) we get that F1 = D4, F2 = D5, and F3 = D6, while F4 = D1, F5 = D2, and F6 = D3. Hence, if S is minimal with a point of degree 3, the birational map φ corresponds over ks to 2 the homaloidal system of quintics passing twice to six points in general position in Pks , three of which are Galois-conjugate and correspond to the closed point of degree 3 on S.

Appendix B. Links of type I, and minimal del Pezzo surfaces of “conic bundle type” Let S be a minimal non-rational del Pezzo surface, which is not deg-rigid. Then S has either degree 8 and a point of degree 2, or degree 4 and a point of degree 1. In both cases, blowing up the given point gives a conic bundle S0 → C over a conic, of degree either 6 or 3, respectively. In this appendix, we would like to show how these two special cases should be thought of, from a derived categorical (or a noncommutative) point of view as conic bundles instead of del Pezzo surfaces. The main point is describing a semiorthogonal decomposition of S whose nonrepresentable components can be seen as the “natural” nonrepresentable components of some conic bundle S0. First of all, let us recall a result of Kuznetsov on the derived category of a conic bundle, as a special case of a quadric fibration (see [71], and [11] for a statement over any field). Proposition B.1. Let π : X → C be a conic bundle over a genus zero curve, A the Azumaya algebra associated to C, and C0 the even Clifford algebra associated to π. Then there is a semiorthogonal decomposition

b b b b D (X) = hD (k), D (k, A), D (C,C0)i, where the two ﬁrst components are the pull-back via π of the natural semiorthogonal decompo- b ∗ sition of D (C). In particular, the ﬁrst one is generated by OX = π OC and the second one by ∗ the local form of π OC (1). In particular, given a conic bundle, we have two natural potentially nonrepresentable com- 1 ponents, one of which is representable in dimension 0 if and only if C ' P .

Del Pezzo of degree 4 with a rational point. Let S be a degree 4 del Pezzo surface with 4 a rational point. Any such surface can be realized in P as an intersection of two quadrics. In b 1 Theorem 5.7, we recalled how AS ' D (P ,C0), where C0 is the Clifford algebra of the quadratic form spanned by the two quadrics. Since S has a point, the fibration has a regular section, so that it can be reduced by hyperbolic splitting (see [11, §1.3]) to a conic bundle with Clifford 0 1 b 1 b 1 0 algebra C0 over P , so that D (P ,C0) ' D (P ,C0). On the other hand, one can blow up the point and obtain a degree 3 surface S0, with a 0 1 structure of conic bundle S → P (see, e.g., [58, Thm. 2.6(i)]). Let us denote by B0 the Clifford algebra of such a conic bundle.

Theorem B.2 ([11, §4]). For S a del Pezzo of degree 4 with a rational point, the OP1 -algebras 0 b 1 C0, C0, and B0 described above are all Morita equivalent. In particular, AS ' D (P ,B0). DEL PEZZO SURFACES 47

Del Pezzo of degree 8 with a degree 2 point. In the case where S is involution surface with a point of degree 2, the nonrepresentable component can also be described by a Clifford 1 algebra over P . Theorem B.3. Let S be a minimal nonrational del Pezzo surface of degree 8, with a closed point x of degree 2. Let S0 → S the blow-up of S along x and π : S0 → C the associated conic 0 0 bundle. Write C = SB(A ) and C0 for the even Clifford algebra of π. Recall the semiorthogonal decomposition b b b b (B.1) D (S) = hD (k), D (k, A), D (k, C0)i, where A is the underlying degree 4 central simple algebra defining S, i.e., S,→ SB(A), and C0 is the even Clifford algebra associated to S. Then A and A0 are Brauer equivalent and there is a semiorthogonal decomposition b 0 b b D (C,C0) = hD (l/k), D (k, C0)i, where l is the residual field of x. Proof. Consider the diagram: S0 σ π φ S / C, where φ is a rational map that is resolved by the blow up σ : S0 → S of x to the conic bundle 0 s ∼ 1 π : S → C. Denote by E the exceptional divisor of σ. Over k , we have that C = Pks and ∼ 1 1 ∗ ∗ S = Pks × Pks and x decomposes into points x1 and x2. Let G = σ O(1, 1) and H = π O(1). As one can check, comparing parts a) and b) of the case K2 = 8 of [58, Thm. 2.6(i)], the s rational map φ is defined, over k , by the linear system OSks (1, 1) − x1 − x2. In particular, s since π resolves φ, we have H = G − L1 − L2 over k , where E = L1 + L2. The semiorthogonal decomposition (B.1) can then be written: b D (Sks ) = hO, O(1, 1), Fi, b where F is the block descending to D (k, C0). Consider the semiorthogonal decompositions of Db(S0) given respectively by the blow up and by the conic bundle formulas: b 0 ∗ (B.2) D (S s ) = hO 0 , O 0 (G), σ F, OL , OL i k Sks Sks 1 2 ∗ Now consider the decomposition (B.2), and mutate σ F to the right with respect to hOL1 , OL2 i, ∗ then mutate hOL , OL i to the left with respect to O 0 (G), then we mutate hO 0 (G), σ Fi to 1 2 Sks Sks the left with respect to its left orthogonal, then we mutate σ∗F to the left with respect to its left orthogonal, then we mutate σ∗F to the right with respect to its right orthogonal to arrive at a decomposition: b 0 ∗ (B.3) D (S s ) = hO 0 (−G + L1 + L2), O 0 , O 0 (G − L1), O 0 (G − L2), σ Fi k Sks Sks Sks Sks where we have used the evaluation sequence and the fact that ω 0 = O 0 (−2G + L1 + L2). Sks Sks Rewriting in terms of H, we have the decomposition b 0 ∗ (B.4) D (S s ) = hO 0 (−H), O 0 , O 0 (H + L2), O 0 (H + L1), σ Fi k Sks Sks Sks Sks ∗ b in which we can identify hO 0 (−H), O 0 i with π D (Cks ). Since O 0 (G) is the mutation Sks Sks Sks of O 0 (−H), and these mutations are defined over k, we conclude that they hO 0 (G)i and Sks Sks b b 0 hO 0 (−H)i descend to equivalent categories over k. It follows that D (k, A) ' D (k, A ), hence Sks by Corollary 1.18, A and A0 are Brauer equivalent. The block hO 0 (H + L2), O 0 (H + L1)i has tilting bundle O 0 (H + L2) ⊕ O 0 (H + L1) = Sks Sks Sks Sks O 0 (H) ⊗ (O 0 (L2) ⊕ O 0 (L1)). There is a vector bundle V on C of rank 2 such that Vks = Sks Sks Sks ⊕2 0 O(H) and there is a vector bundle W on S of rank 2 such that Wks = O(L1) ⊕ O(L2). Then ∗ π V ⊗ W is a tilting bundle for a category base changing to the block hO 0 (H + L2), O 0 (H + Sks Sks 48 ASHER AUEL AND MARCELLO BERNARDARA

b L1)i. It follows that the later descends to a category equivalent to D (k(x)/k, A). Since k(x) is the residue field of a point of S ⊂ SB(A), we have that k(x) splits the algebra A. When b b ∗ b 0 b D (k(x)/k, A) ' D (k(x)/k). In conclusion, we get a decomposition π D (C,C0) = hD (l/k), Fi b and we recall that F ' D (k, C0). Appendix C. Links of type II between conic bundles of degree 8 In this section, we consider conic bundles of degree 8 and study their semiorthogonal decompositions under links of type II and IV. The upshot is to show that the Griffiths–Kuznetsov component is well defined in these cases, which include in particular all involution surfaces of Picard rank 2. Let us first consider a conic bundle π : S → C over a Severi–Brauer curve C = SB(A). Such a conic bundle has an associated Clifford algebra C0, a locally free sheaf over C. Kuznetsov provides a semiorthogonal decomposition: b b b D (S) = hD (C), D (C, C0)i, and shows that there is a root stack structure Cb, obtained by the natural Z/2Z-action on points of C where the fiber is degenerate, and a Brauer class β in Br(C) such that C0 pulls back to Cb to an Azumaya algebra with class β. Recall that a conic bundle over C has degree 8 − r, where r is the number of degenerate fibers. If S has degree 8, then it has no degenerate fibers and hence Cb =∼ C since the root stack structure is trivial. Recall moreover that, denoting by α ∈ Br(k) the class of A, there is a semiorthogonal decomposition Db(C) = hDb(k), Db(k, α)i. We finally obtain a semiorthogonal decomposition (C.1) Db(S) = hDb(k), Db(k, α), Db(k, β), Db(k, α ⊗ β)i. We can show that the nontrivial components of this decomposition are a birational invariant, which allows us to conclude that the Griffiths–Kuznetsov component is well defined.

Proposition C.1. Suppose that S → C is a conic bundle of degree 8 and that S1 99K S is a birational map. Then there is a semiorthogonal decomposition b b 0 b AS0 = hT, D (k, α), D (k, α ), D (k, α ⊗ β)i 1 b Remark C.2. Notice that if α = 0 (i.e. C = P ), we can include D (k, α) in T, and similarly for β = 0 (i.e. π has a section), we can include Db(k, β) in T s Proof. Notice that, over k , the conic bundle Sks is isomorphic to a Hirzebruch surface Fn, that 1 1 is, a P -bundle π : Sks → Pks . One can check that the semiorthogonal decomposition (C.1) base-changes to the semiorthogonal decomposition b s (C.2) D (Sk ) = hOSks , OSks (F ), OSks (Σ), OSks (Σ + F )i, where we denoted by F and Σ the fiber and the section of π respectively. Suppose first that S1 is minimal, so that S can be decomposed in a series of links of type either II or IV (a link of type III is a blow-down, so S1 is not minimal). Let us first consider links of type II, which are described by Iskovskikh [58, Thm. 2;6 (ii)]. Let π : S → C be a conic bundle of degree 8, and σ τ S o X / S0

=∼ CCo 0 where σ : X → S and τ : X → S0 are blow-ups in a point x and x0 respectively of the same degree d, and S0 → C0 is a conic bundle. We denote by E and E0 the exceptional divisors of the blow ups. We ﬁrst notice that S0 has also degree 8 and that C0 ' C. The degree d of the point can be either 1, 2, or 4. It is easy to check that, over ks , the link is just a composition of d elementary transformations of the Hirzebruch surface Sks . In particular, if there is a point of degree 1, then we obtain a birational map between S and a quadric with a rational point, so that S is already a Hirzebruch surface and there is nothing to prove. So we assume that DEL PEZZO SURFACES 49

s d is either 2 or 4. Moreover, as one can check over k , we have −KS = 2Σ − (n − 2)F and 0 0 −KS0 = 2Σ − (n + d − 2)F . In the Picard group of Xks we have the following relations (we omit the pull-back notation): (C.3) F = F 0, (C.4) E = dΣ0 − E0, 0 0 (C.5) −KS = −KS0 + dF + 2E , (C.6) Σ = Σ0 − E0, where the last equality, is obtained combining the first and the third one. In particular, we 0 0 b obtain an identification O(F ) with O(F ) and that the the equivalence ⊗O(−E ) of D (Xks ) sends the exceptional line bundle O(Σ) to the exceptional line bundle O(Σ0) and the exceptional line bundle O(Σ + F ) to the exceptional line bundle O(Σ0 + F 0). We notice that, since E0 is defined over k, the latter equivalence is defined over k as well. From this we obtain equivalences Db(k, α) ' Db(k, α0) (the identity), Db(k, β) ' Db(k, β0) and Db(k, α ⊗ β) ' Db(k, α0 ⊗ β0) (induced by ⊗O(−E0)). Let us now consider links of type IV. Thanks to Iskovskikh, this is possible only in the cases where S = S0 = C × C0 is the product of two Severi–Brauer curves and the conic bundle structures are given by the two projections. Let α and α0 in Br(k) be the classes of C and C0 respectively. Then there are natural decompositions: b b b b b 0 b 0 D (C) = hD (k), D (k, α)i D (C, C0) = hD (k, α ), D (α ⊗ α )i (C.7) b 0 b b 0 b 0 b b 0 D (C ) = hD (k), D (k, α )i D (C, C0) = hD (k, α), D (α ⊗ α)i It follows that the decompositions (C.1) obtained by the conic bundle structures are composed by equivalent categories. This proves the Proposition for S1 minimal. If S1 is not minimal, then just consider a minimal model S1 → S0 and use the blow-up formula. We finally classify birational equivalent involution surfaces based on algebraic methods. This can be seen as the algebraic interpretation of the classification of all type IV links for such surfaces. Proposition C.3. Let S and S0 be nonrational involution varieties over a field k. Then S 0 and S are k-birational if and only if either they are isomorphic or S = SB(B1) × SB(B2) and 0 0 0 S = SB(B1) × SB(B2) and the Klein four subgroups of the Brauer group generated by Bi and 0 Bi coincide and consist solely of quaternion algebras. Proof. Let S and S0 are be k-birational involution varieties associated to quadratic pairs (A, σ) and (A0, σ0). The only case not ruled out by Proposition A.2 is that of S (hence S0) of index 2, trivial discriminant, and with (at least) (A, σ) nonsplit. By the classification of degree 4 algebras with orthogonal involution of trivial discriminant [67, §15.B], (A, σ) is isomorphic to + + − − ± ± (C0 , τ0 ) ⊗ (C0 , τ0 ), where τ0 is the standard involution on the quaternion algebra C0 , and similarly for (A0, σ0). By (A.2) and the considerations in the proof of Proposition A.2, the Klein ± 0 ± four subgroups of Br(k) generated by C0 and C0 coincide and consist solely of quaternion algebras (since A and A0 have index at most 2). 0 0 Now we verify that if B1,B2 and B1,B2 are Brauer classes generating the same Klein four 0 0 subgroup consisting of classes of index at most 2, then S = SB(B1)×SB(B2) and S = SB(B1)× 0 0 0 SB(B2) are k-birational. If the unordered pairs (B1,B2) and (B1,B2) are isomorphic then ∼ 0 0 0 S = S . Otherwise, without loss of generality, we can write B1 = B1 and B2 = H, where ∼ B1 ⊗ B2 = M2(H). However, since [H] = [B1] + [B2] ∈ Br(k) and ker(Br(k) → Br(SB(B1))) = 0 h[B1]i, the images of [H] and [B2] coincide in Br(k(SB(B1))), and thus we conclude that k(S ) = ∼ 0 k(SB(B1)) ⊗ k(SB(H)) = k(SB(B1)) ⊗ k(SB(B2)) = k(S) so that S and S are k-birational. 1 As a special case, this result says that the involution surfaces C×C and P ×C are k-birational for a any Severi–Brauer curve C over k. 50 ASHER AUEL AND MARCELLO BERNARDARA

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Department of Mathematics, Yale University, 10 Hillhouse Avenue, New Haven, CT 06511 E-mail address: [email protected]

Institut de Mathematiques´ de Toulouse, Universite´ Paul Sabatier, 118 route de Narbonne, 31062 Toulouse Cedex 9, France E-mail address: [email protected]